THE LONG RANGE DISPERSION OF
RADIOACTIVE PARTICULATES
by
Joshua Michael Aaron Ryder
Wurman
S.B. Massachusetts Institute of Technology, 1982
Submi tted to the Department of
Meteorolog y and Physical Oceanography
in Part ial Fulfillment of the
Requir ements of the Degree of
MASTER OF SCIENCE IN METEOROLOGY
at the
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
June 1982
@Joshua Michael
Aaron Ryder Wurman
1982
The author hereby grants to the Massachusetts
Institute of Technology permission to reproduce
and distribute publicly copies of this thesis
document in whole or in part.
Signature of Author
Department of Meteorology and Physical Oceanography
18 June 1982
Certified by_,
Dr.
'I,
Raymond Pierrehumbert
Thesis Supervisor
Accepted by
Chairman, Department of
Dr'. Peter Stone
Meteorology and Physical Oceanography
LnaoWTHRTAWR
FROM-1982
M IT LiB RARE s
PAGE 2
THE LONG RANGE DISPERSION OF
RADIOACTIVE PARTICULATES
by
Joshua Michael Aaron Ryder Wurman
S.B. Massachusetts Institute of Technology, 1982
Submitted to the Department of
Meteorology and Physical Oceanography
in Partial Fulfillment of the
Requirements of the Degree of
Master of Science in Meteorology
ABSTRACT
Particl e advection in the upper troposphere and lower
mo deled
using the Arakawa scheme to
stratosphere
was
approximate the Jacobia n of the height of constant pressure
and the conce ntration field in the concentration
field
The scheme was tested
with
three
advection e quation.
The
different initializati ons of particle dis-tributions.
Wakes of
numerical properties of the scheme were studied.
alternating sign produc ed by the scheme to the lee side of the
be
unacceptable
particle cloud were found to
and
a
modification to the sc heme was proposed. The modified scheme
The
three
initializations.
same
was tested with the
modification was foun d to eliminate the wakes with only a
small result ant numeric al diffusion. A crude model for the
deposition of particl es to the ground was added to the
The resultant scheme was tested
and
advection scheme.
produced a realistic fallout pattern. Horizontal diffusion
was neglecte in all the tests. The tests used an idealized
repre sen ting purely zonal flow. The model is
height fiel
syn optic height fields, real diffusion
adaptable to real
coefficients and more realistic deposition schemes.
Thesis Supervisor:
Dr. Raymond Pierrehumbert
Title:
Assistant Professor of Meteorology
PAGE 3
ACKNOWLEDGMENTS
This research
Undergraduate
was
Research
thank the people who run
project
may
never
have
begun
in
January,
Opportunities
this
program,
started.
1981
Program.
without
the
past
eighteen months.
computer
facility
running
wish
which
to
this
invaluable
I wish to thank Ken Morey
whose assistance and patience with me and efforts
the
I
the
I wish to thank my thesis
advisor, Raymond Pierrehumbert, whose guidance was
during
under
permitted
at
keeping
the computational
phase of this research to occur.
Finally, I wish to thank my parents, who instilled in me
the greatest of all gifts, the desire to learn.
PAGE 4
TABLE OF CONTENTS
Title page
1
Abstract
2
Acknowledgments
3
Table of Contents
4
List of Figures
8
List of Tables
19
Definition of Symbols
20
I. Introduction
23
II. Derivation of exact equations
30
III. Derivation of finite difference forms.
39
IV.
46
Trials
A. Description of computational method and data
46
B. Individual trials
57
1. Unsmoothed trialsa. One point initialization
57
57
i. Description of the initialization
57
ii. Diagrams of the advection
57
iii. Description of the advection
60
iv. Cross-sections
62
v. Mass in the forward positive area
71
vi. Velocity of the forward positive area
73
vii. Conservation of mass
74
viii. Conservation of squared mass
76
ix.
Discussion
b. Nine point init'ialization
78
78
PAGE 5
i. Description of the initialization
78
ii. Diagrams of the advection
79
iii. Description of the advection
81
iv. Cross-sections
83
v. Mass in the forward positive area
90
vi. Velocity of the forward positive area
91
vii. Conservation of mass
92
viii. Conservation of squared mass
94
c. Sixty-nine point initialization
96
i. Description of the initialization
96
ii. Diagrams of the advection
97
iii. Description of the advection
100
iv. Cross-sections
102
v. Mass in the forward positive area
108
vi. Velocity of the forward positive area
109
vii. Conservation Of mass
109
viii. Conservation of squared mass
111
2. Smoothed advection
114
a. Discussion
114
b. One point initialization
114
i. Description of initialization
114
ii. Diagrams of the advection
115
iii. Description of the advection
118
iv. Cross-sections
119
v. Velocity of the cloud
124
vi. Conservation of mass
127
vii. Conservation of squared mass
129
PAGE 6
v iii. Discussion
c. Nine-point initialization
134
134
i . Description of initialization
134
ii. Diagrams of the advection
134
iii.
137
Description of the advection
iv. Cross-sections
137
. Velocity of the cloud
142
vi. Conservation of mass
143
vii. Conservation of squared mass
145
d. Sixty-nine point initialization
147
i. Description of initialization
147
ii. Diagrams of the advection
148
iii.
150
v
Description of the advection
iv. Cross-sections
150
v . Velocity of the cloud
155
i. Conservation of mass
156
v
v ii. Conservation of squared mass
3. Trial with deposition
158
161
a. Discussion
161
b. Description of deposition schem e
161
c. Description of initialization
163
d. Deposition parameters
163
e. Diagrams of the advection
164
f. Description of the advection
167
g. Cross-sections
168
h. Conservation of mass
173
i.
Conservation of squared mass
175
PAGE 7
j.
Diagrams of deposition
177
k. Description of deposition
181
1. Cross-sections through the deposition pattern
181
V. Conclusions
186
A. Unsmoothed model
186
B. Smoothed model
188
C. Deposition case
191
D. Summary
192
References
194
Appendix A
197
Appendix B
200
PAGE 8
LIST OF FIGURES
figure
page
1.
Wakes in Orzag's advection trials
44
2.
Longitudinal gridpoint spacing
48
3.
Height fie ld used in the trials
52
4.
Wind speed versus latitude in m s-1
55
5.
Wind speed versus latitude in gridpoints s-1
56
6a.
One point in iti al ization unsmoothed:
advection pattern at t=0000
6b.
One point initial ization unsmoothed:
advection pattern at t=0001
6c.
58
One point initial ization unsmoothed:
advection pattern at t=0010
6f.
58
One point initial ization unsmoothed:
advection pattern at t=0003
6e.
57
One point initial ization unsmoothed:
advection pattern at t=0002
6d.
57
58
One point initial ization unsmoothed:
advection pattern at t=0100
6g.
One point initial ization unsmoothed:
advection pattern at t=1440
7a.
One point initial ization unsmoothed:
cross-section at t=0020
7b.
One point initialization unsmoothed:
cross-section at t=0050
64
PAGE 9
figure
7c.
One point initialization unsmoothed:
cross-section at t=0100
7d.
65
One point ini tial ization unsmoothed:
cross-sect ion at t=0200
7e.
page
66
One point ini tial ization unsmoothed:
cross-section at t=0300
7f.
One point ini tial ization unsmoothed:
cross-section at t=0600
7g.
One point ini tial ization unsmoothed:
cross-section at t=1440
8.
69
One point ini tial ization unsmoothed:
normalized mass
9.
68
75
One point ini tialization unsmoothed:
normalized squared mass
77
10a. Nine point initial ization unsmoothed:
advection pat tern at t=0000
79
10b. Nine point initial ization unsmoothed:
advection pat tern at t=0001
79
10c. Nine point initial ization unsmoothed:
advection pat tern at t=0002
79
10d. Nine point initial ization unsmoothed:
advection pat tern at t=0003
80
PAGE 10
figure
10e.
page
Nine point initialization unsmoothed:
advection pattern at t=0010
10f.
Nine point initialization unsmoothed:
advection pattern at t=01 00
11a.
Nine point initialization unsmoothed:
cross-section at t=0060
11b.
Nine point initialization unsmoothed:
cross-section at t=0100
11c.
Nine point initialization unsmoothed:
cross-section at t=0200
11d.
86
Nine point initialization unsmoothed:
cross-section at t=0300
11e.
85
87
Nine point initialization unsmoothed:
cross-section at t=0600
11f.
Nine point initialization unsmoothed:
cross-section at t=1440
12.
Nine point initialization unsmoothed:
normal ized mass
13.
Nine point initialization unsmoothed:
normalized squared mass
14a.
Sixty-nine point initiali zation unsmoothed:
advection pattern at t=0000
98
PAGE 11
figure
page
14b. Sixty-nine point initialization unsmoothed:
advection pattern at t=0001
98
14c. Sixty-nine point initialization unsmoothed:
advection pattern at t=0002
98
14d. Sixty-nine point initialization unsmoothed:
advection pattern at t=0003
99
14e. Sixty-nine point initialization unsmoothed:
advection pattern at t=0010
99
14f. Sixty-nine point initialization unsmoothed:
advection pattern at
t=0100
99
15a. Sixty-nine point initialization unsmoothed:
cross-section at t=0050
103
15b. Sixty-nine point initial ization unsmoothed:
cross-section at t=0180
104
15c. Sixty-nine point initial ization unsmoothed:
cross-section at t=0300
105
15d. Sixty-nine point initialization unsmoothed:
cross-section at t=0600
106
15e. Sixty-nine point initialization unsmoothed:
cross-section at t=1440
16.
107
Sixty-nine point initialization unsmoothed:
normalized mass
110
PAGE 12
figure
17.
page
Sixty-nine point initialization unsmoothed:
normalized squared mass
112
18a. One point initialization smoothed:
advection pattern at t=0001
115
18b. One point initial ization smoothed:
advection pattern at t=0002
116
18c. One point initial ization smoothed:
advection pattern at t=0003
116
18d. One point initial ization smoothed:
advection pattern at t=0010
116
18e. One point initial ization smoothed:
advection pattern at t=0100
116
18f. One point initial ization smoothed:
advection pattern at t=0300
117
18g. One point initial ization smoothed:
advection pattern at t=0600
117
18h. One point initial ization smoothed:
advection pattern at t=1440
117
19a. One point initial ization smoothed:
cross-sect ion at t=0100
120
19b. One point initial ization smoothed:
cross-section at t=0300
121
PAGE 13
figure
page
19c. One point initialization smoothed:
cross-section at t=0600
122
19d. One point initialization smoothed:
cross-section at t=1440
123
20.
Cause of centroid lag in smoothed advection
126
21.
One point initialization smoothed:
normalized mass
22.
128
One point initialization smoothed:
normalized squared mass
130
23a. Cause of squared mass variation:
unsmoothed advection
131
23b. Cause of squared mass variation:
smoothed advection
132
24a. Nine point initialization smoothed:
advection pattern at t=0001
135
24b. Nine point initialization smoothed:
advection pattern at t=0002
135
24c. Nine point initialization smoothed:
advection pattern at t=0003
135
24d. Nine point initialization smoothed:
advection pattern at t=0010
135
24e. Nine point initialization smoothed:
advection pattern at t=0100
136
24f. Nine point initialization smoothed:
advection patter'n at t=0300
136
PAGE 14
figure
page
24g. Nine point initial ization smoothed:
advection pattern at t=0600
136
24h. Nine point initial ization smoothed:
advection pattern at t=1440
136
25a. Nine point initialization smoot hed:
cross-secti on at t=0100
138
25b. Nine point initialization smoot hed:
cross-secti on at t=0300
139
25c. Nine point initialization smoot hed:
cross-secti on.at t=0600
140
25d. Nine point initialization smoot hed:
cross-secti on at t=1440
26.
Nine point
initialization smoot hed:
normalized mass
27.
141
'
144
Nine point initialization smoot hed:
normalized squared mass
146
28a. Sixty-nine point initialization smoothed:
advection pattern at t=0001
148
28b. Sixty-nine point initialization smoothed:
advection pattern at t=0002
148
PAGE 15
figure
page
28c. Sixty-nine point initialization smoothed:
148
advection patter n at t=0003
28d. Sixty-nine point initialization smoothed:
149
advection patter n at t=0010
28e. Sixty-nine point initialization smoothed:
149
advection patter n at t=0100
28f. Sixty-nine point initialization smoothed:
149
advection patter n at t=0300
28g. Sixty-nine point initialization smoothed:
149
advection patter n at t=0600
28h. Sixty-nine point initialization smoothed:
149
advection patter n at t=1440
29a. Sixty-nine point initialization
smoothed:
cross-sect ion at t=0100
151
29b. Sixty-nine point initialization smoothed:
cross-sect ion at t=0300
152
29c. Sixty-nine point initialization smoothed:
cross-sect ion at t=0600
153
29d. Sixty-nine point initialization smoothed:
cross-sect ion at t=1440
30.
154
Sixty-nine point initialization smoothed:
normalized mass
157
PAGE 16
figure
31.
32.
page
Sixty-nine point initialization smoothed:
normalized squared mass
159
Process of deposition
162
33a. Trial with deposition:
advection pattern at t=0001
164
33b. Trial with deposition:
advection pattern at t=0002
165
33c. Trial with deposition:
advection pattern at t=0003
165
33d. Trial with deposition:
advection pattern at t=0010
165
33e. Trial with deposition:
advection pattern at t=0100
165
33f. Trial with deposition:
advection pattern at t=0300
166
33g. Trial with deposition:
advection pattern at t=0600
166
33h. Trial with deposition:
advection pattern at t=1440
166
34a. Trial with deposition:
cross-section at t=0500
169
PAGE 17
figure
page
34b. Trial with deposition:
cross-section at t=0600
170
34c. Trial with deposition:
cross-section at t=1000
171
34d. Trial with deposition:
cross-section at t=1440
172
35. Trial with deposition:
normalized mass
174
36. Trial with deposition:
normalized squared mass
176
37a. Trial with deposition:
deposition pattern at t=0001
177
37b. Trial with deposition:
deposition pattern at t=0002
177
37c. Trial with deposition:
deposition pattern at t=0003
177
37d. Trial with deposition:
deposition pattern at t=0010
177
37e. Trial with deposition:
deposition pattern at t=0100
178
37f. Trial with deposition:
deposition pattern at t=0300
178
PAGE 18
figure
page
37g. Trial with deposition:
deposition pattern at t=0600
178
37h. Trial with deposition:
deposition pattern at t=1000
179
37i. Trial with deposition:
deposition pattern at t=1440
180
38a. Trial with deposition:
deposition cross-sections at t=0010
182
38b. Trial with deposition:
deposition cross-sections at t=0100
183
38c. Trial with deposition:
39.
deposition cross-sections at t=1000
184
Wake formation regions
201
PAGE 19
LIST OF TABLES
table
page
1. One point initialization unsmoothed:
magnitudes of wake and the forward positive area
61
2. One point initialization unsmoothed:
excess concentration in the forward positive area
71
3. Nine point initialization unsmoothed:
magnitudes of wake and the forward positive area
82
4. Nine point initialization unsmoothed:
excess concentration in the forward positive area
91
5. Sixty-nine point initialization unsmoothed:
magnitudes of wake and the forward positive area
101
6. Sixty-nine point initialization unsmoothed.
excess concentration in the forward positive area
108
7. One point initialization smoothed:
lag in the centroid
125
8. Nine point initialization smoothed:
lag in the centroid
142
9. Sixty-nine point initialization smoothed:
lag in the centroid
155
PAGE 20
DEFINITION OF SYMBOLS
symbol
description
page first used
a
concentration in primary grid box
126
b
concentration in secondary grid box
126
cc
used to calculate smoothed height field
198
C
concentration
30
cos
cosine
33
d
differential operator
30
a
partial differential operator
30
D
total differential operator
30
D,
vertical distance scale
30
DE
diffusion constant
27
dd
used to calculAte smoothed height field
198
ee
used to calculate smoothed height field
198
f
coriolis parameter
ff
used to calculate smoothed height field
9
gravitational acceleration of the Earth
99
used to calculate smoothed height field
gp
gridpoint
H
vertical depth of the cloud of particles
163
hh
used to calculate smoothed height field
199
J
Jacobian
35
unit vector perpendicular to the Earth's surface
41
kilograms
49
kg
32
199
35
199
53
PAGE 21
symbol
description
page first used
L
horizontal distance scale
ln
longitudinal grid coordinate
198
1t
latitudinal grid coordinate
198
m
meters
27
mb
millibars
49
om
angular rotation rate of the Earth
34
0
of order
39
p
density of air
32.
P
pressure
32
Pa
pascals
49
ph
zonaT distance coordinate in natural system
33
r
gas constant for air
35
Re
radial distance from the center of the Earth
33
RS
Rossby number.
30
R0
radius of Earth
37
RL
radius of curvature of wind flow
32
s
seconds
27
sin
sine
34
t
time
28
T
temperature
35
th
meridional distance coordinate in natural system
33
tstp
timestep
53
u
zonal velocity
30
u
zonal geostrophic wind velocity
32
Li
zonal velocity
30
30
PAGE 22
symbol
description
page first used
v
meridional velocity, velocity
28
v
meridional geostrophic wind velocity
32
V
meridional velocity, velocity
32
V
geostrophic wind velocity
32
VC
cyclostrophic wind velocity
32
w
vertical velocity
30
x
zonal distance coordinate in cartesian system
30
y
meridional distance coordinate in cartesian system 30
z
vertical distance coordinate in cartesian system
30
vorticity
41
A
incremental change
47
*
multiplication operator
34
X
cross product operator
41
.
dot product operator
41
V
del operator
41
V
Laplacian operator
36
v
mean terminal velocity
The units used in this paper
specified.
Units
are
expressed
161
are
MKS
unless
in exponent
otherwise
form
velocity is expressed as a certain number of m s-1 )
( e.g.
PAGE 23
I. Introduction
It is likely that a large scale military conflict between
the
United
States
and
the
exchange of nuclear weapons.
devices
is
orders
of
Soviet
The
Union would involve the
explosive
magnitude
conventional, non-nuclear, weapons.
greater
The blast
force
of
than
these
that
damage
of
caused
by the detonation of these devices is, of course, much greater
than that from a conventional explosion.
nuclear
detonations
extend
far
beyond
caused by the force of the explosion.
is the
study
of
a phenomenon
But the
effects
of
the. initial damage
The focus of this paper
unique
to nuclear weapons:
radioactive fallout.
Radioactive fallout *iscomprised of radioactive particles
from
the
bomb
itself and from debris that is exposed to the
The amount and nature of
nuclear blast.
radioactive
fallout
is dependent on the nature of the bomb and the altitude of the
detonation above the ground.
Earth's
surface
there
In an explosion
above
the
is little solid material present from
which fallout can be created.
the
far
When the explosion occurs
near
ground, however, a large amount of the surface is exposed
to the
blast
and
thrown
into
the
atmosphere,
become
In the event of a war, weapons that are targeted at
fallout.
strengthened missile silos and other military
likely,
to
be
detonated
targets
would,
near ground level in order to maximize
PAGE 24
their destructive force.
debris
that
thrown
into
the
of
much
atmosphere
will
the
become
The effects of the resultant radioactivi ty
radioactive.
these
is
During such an event,
particles
eventually
fall
when
to the ground are commonly
known and can be disastrous.
The fallout, just described, is divided
types,
distinguished
radioactivity
consists
of
to
by
large
included
in
this
the
gr ound.
radioactive
ground rapidly and near
the
Near
particles
sit e
category
of
the
ground.
basic
range
that
fallout
fall to the
explosion.
Also
are small particles that are not
thrown far into the atmosphere and, thus,
the
two
time that is necessary for the
the
reach
into
settle
rapidly
to
This near range fal lout is due to particles that
have only a short resioence time in the atmosphere and only
short
exposure
to
the
transpo rt effects of the wind.
paper deals with the long range
nuclear explosions.
at
the
surface.
1ower
the
by
surface.
by
large
During
time, high winds at that altitude cause the particles to
may
circuit
The
particulate
the globe many times before nearly all of
the particles fall and the cloud diffuses and -loses
Most
caused
stratosphere
advect away from the site of the explosion.
cloud
is
These particles, often very small,
require from hours to years to fa11 to
this
that
This
This fallout consists of particles thrown
into the upper troposphere and
blasts
fallout
a
of
the
destructive
cloud's first day or tivo of
identity.
fallout, however, falls during the
advection
on
its
first c
c ir cu it
PAGE 25
around the Earth.
Studies of the movement
fallout
from them have been made by other
and Zagurek (1981)
be
calculated the fall out
silos in Uta h, Nev ada,
prediction
Texas,
and New
WSEG 10
Their
f or
predicted
S tudy,
a
scale of mor tal ity
on
the
while
"typ ical
and morbidity that
MX missile silos.
This
a
The
scheme
was
magnitude
of
the
eddy
displacements
effects
indicated the
" day,
d occu r in
such
an
area in th e West all the
A study of particl e trajectories
by
Kao
The subject of their
and
Henderson
paper
was
the
diffusivity and statis tics about the
after
Tanaevsky and Blanchet (1966)
very
long
t ime
periods.
studied the fallout from nuclear
detonations in the atmosphere over
in
woul
the
study modeled individ ual particl e trajectories
in a Lagrangian manner.
Zemlya
used
Harmf ul lev els of fallout
in variable wind fields was conducted
particle
They
indi cating
only
non-existent,
the Atlantic cOast.
(1970).
would
Emergency Management
Model
Fallout
were predicted to fall from the blast
to
that
Wit
a wind f ield that was descr ibed as a "'typical day'
in March".
way
amounts
Mex ico.
utilized by the Fede ral
model
Agency calle d the
tested on
investigators.
from an attack on the, once, proposed MX missile
expected
attack
of these clouds and the resultant
the Soviet Union.
Semipalatins k
and
Novaya
They found it difficult to plot
cloud trajectories, especially in regions
of
difluent
flow.
Instead,
probability"
which
they
plotted
a
"line
of
represented the front along which the first fallout was likely
PAGE 26
to occur.
Due to the indeterminate prediction of the location
of the cloud, no attempt was made to predict
fallout
amounts
at specific locations.
In this paper, the
during
its
first
movement
of
the
particulate
day of advection is modeled in an Eulerian
scheme involving the fields of concentration and
pressure
heights.
directly
related
cloud
The
to
of
constant
height field of constant pressure is
the
wind
field
approximation to the momentum equation.
by
the
geostrophic
The advection term in
the rate of concentration change equation is written in terms
Of
a Jacobian
of
the concentration and height fields.
Jacobian is approximated by
a
scheme
developed
by
The
Arakawa
(1966).
The Arakawa scheme has advantages and drawbacks that
discussed,
at
length, in the text.
are
Its basic advantages are
designed to be the conservation of mass and the suppression of
spurious numerical diffusion.
In the trials presented in this
paper, the scheme conserves mass very closely.
designed
to
fairly well.
masses
in all
grid
boxes,
squares
of
model produced
series
of
is also
The drawback to this scheme was that,
in order to conserve the quantity just described, the
the
quantity
measure resistance to numeric diffusion, the sum
of the squares of the
conserved
The
sum
of
the masses in all portions of the cloud, the
unrealistic
wakes,
results.
The
scheme
caused
a
alternating in sign, to develop behind and
PAGE 27
around the cloud.
These wakes eventually dominated the
and obscured any useful results.
included,
actually
dimensions.
The
expanded
The cloud, if the wakes were
very
rapidly
While
continental
negative
and
positive
the wakes cause the pure Arakawa scheme to not
be usable in this context, a scheme is
eliminates
to
sum of the squares quantity is conserved at
the expense of creating the series of
wakes.
cloud
the
wakes
and
presented
here
produces good results.
which
The small
resultant degree of numerical diffusion, that is noticeable by
inspection
of
the
cloud
and
by the drop in the sum of the
squares quantity, is acceptable and
the
cloud
appears
very
realistic.
Real turbulent diffusion is ignored in the tests
modified
scheme
that
are
presented in this paper.
done so that the properties of the
studied
without
advection
scheme
the
ThiS is
can
be
the complications introduced by the physical
diffusion of the cloud.
The advection pattern is not
significantly
omission.
by
of
this
Davidson,
changed
et. al. (1966)
determined that the range of horizontal diffusion coefficients
in
the
troposphere
and
stratosphere
D = 10000 m+2 s-1 and D = 100000 m+2 s-1.
the
very
long
term
was
In their
between
model
of
di ffusion of Tungsten-185,
a value of D
= 40000 m+2 s-1 produced the best agreement with the observed
distribution
of
the
element months after its injection into
the atmosphere by nuclear detonations.
the
distribution
Dyer
(1966)
studied
of 'vol canic dust in the Southern Hemisphere
PAGE 28
during the two years following the eruption of Mount
1963.
He
determined that the con centra tion of dust found at
Aspendale, Victoria, Australia coul d best
value
of
the
be explained if
diffusion constant was Di = 40000 m+2 s-1.
amounts found at the South Pole wer e best
of
D;= 140000 m+2 s-1.
The
Even with the lar gest of these values,
one
The
day.
advective
diffusive length scale,
, assuming D,= 100000 m+2 s-1 and t= 100000 s, is of
L ~ ViD
order
after
scale
the
explained by a value
the diffusive length scale is much small e r than the
length
Bali
100000M.
The deposition pattern has a.length scale in
the direction of the wind vel ocity that is of order L ~ v * t.
If
a
wind
speed..of
10 m s-1
and
th e
same time scale is
assumed, then this distance is of order 1 ,000,000 m.
long
axis of the deposition pattern will
affected by the
deposition
neglect
of
diffusion.
Thus the
not be significantly
The
width
pattern is much 1ess than its length.
of
the
But, if, as
will be shown in the trials, the minimum stable cloud size
is
of order five gridpoints or 100,000 m, then the diffusion will
only double this scale.
Thus ,
the
bulk
properties
of
the
deposition remain the same ev en with the neglect of diffusion.
This is especially true in
diffusion
that
is
the
present
trials
helps
where
produce
the
numerical
more
realistic
results.
An idealized height field is used in order
properties of advection scheme
to
test
the
The purpose of this study was
not to make actual predictions of real fallout patterns.
The
PAGE 29
scheme
described
in the text is, however, fully adaptable to
real synoptic situations.
The deposition scheme used in this paper is
This
scheme
The
crude.
is included to indicate the general intensity of,
fallout that could be expected
cloud.
very
assumptions
of
to
no
fall
from
vertical
a
particulate
diffusion
and
a
mono-disperse particle size distribution are made and they are
clearly not realistic.
This paper is intended to provide some
basic
groundwork
towards the prediction of actual distributions under realistic
and varying atmospheric conditions.
The
assumptions
of
an
idealized wind field, zero horizontal or vertical diffusion, a
crude deposition scheme, and the neglect of other effects such
as
the
effects
radioactive decay of the particles and the scavenging
of
atmosphere,
precipitation
cause
in
the
lower
portions
of
the
the trials presented here to be of little,
actual predictive, significance.
The techniques
include
these,
and
described
possibly
here
other,
should be applied to real wind fields.
better
understanding
of
the
explosions will. be achieved.
long
should
be
effects
expanded
to
and the model
Once this is
done,
a
range effects of nuclear
PAGE 30
II. Derivation of exact equations
In order to predict the concentration
particular
location
of
at
a
at a particular time, it is necessary to
know both the concentration at an ini tial
able
particulates
time
and
then
be
to integrate the time rate of change of concentration at
that location.
The total time
rate
of
change
of
particle
concentration can be expressed as
DC
dC
Dt
ac
dC
u-- +
ax
dt
V--
ay
ac
+ w--
(1)
az
where C is the particle concentration, and u, v, and w are the
flow
velocities
in the x, y, and z directions respectively,
and the particles
are
assumed
to
be
moving
at
the
same
velocity as the wind.
Scaling arguments can
be
used
to
simplify
(1).
The
verti cal motions associated wi th synoptic scale flow are quite
smal 1
in
the
atmosphere.
The
followi ng
scaling
of
the
verti cal wind (Pedlosky, 1979) shows this explicitly.
w
U [
[
DV ]
---
(2)
] R
L ]
where U is a typical horizontal vel ocity,
of
vertical
distance' ,
L
D. is a typical scale
is a typical scale of horizontal
PAGE 31
distance, and R is the Rossby number.
The
ratio
of
the
vertical
advection
term
to
the
horizontal advection terms is then:
[ D, R,][
dC
w --
U[ ---
dC
u --
[
U[
C
]
]( --- ]
dz
[
L
]( D, ]
----- ~-------------= R
[
ax
(3)
C
-
L
Thus, if the Rossby number is small, then the last term in (1)
can be neglected with respect to the other two.
The
advective
velocities.
For
terms
in
(1) explicitly
directly.
certain
standard
advection.
and
lower
stratospheric
It is more useful to use the height field of
pressure
values
to
calculate
particle
In order to calculate the particle advection from
this height data it is first necessary to transform
pressure
wind
calculation purposes, it is inconvenient to
utilize observed upper tropospheric
winds
contain
coordinates.
used in order to do this.
are (Holton,
1979):
(1)
into
The horizontal momentum equations are
The simplified
momentum
equations
PAGE 32
du
--
1
-
v
f
=
-
dt
p
aP
(4)
ax
1 dp
dv
--
-
+fu
=
-
-
(5).
p ay
dt
where f is the coriolis parameter, p is
air,
and
P is the local pressure.
the
then the first terms can both be
resul ts,
fter
terms
of
the
If the acceleration terms
are small,
rearranging
density
are
dropped
(6a)
and
and
the
(6b), the
geostrophi : approximation.
aP
1
(6a)
p f dx
aP
-1
a
(6b)
p f dy
This approximation is accurate to order R,, the Rossby
The
greatest
deviations
from
geostrophy
will occur in the
highest curvature regions of mid-latitude jets.
from
the
more
accurate,
but
number.
The deviation
more computationally involved-
gradient, or cyclostrophic, wind is:
VC
-
V
1+
V
----fR
(7)
PAGE 33
where RL is the radius of curvature of the flow pattern.
cyclonic flow and negative in anticyclonic flow.
in
positive
R. is
ten
than
more
rarely
is
The departure from geostrophy
or
percent in middle and high latitudes, but can be great
twenty
geostrophic
the
with
Thus,
in the tropics.
approximation,
study confines itself to the examination of advection in
this
the middle and high latitudes.
The substitution of (6a) and (6b) into (1) gives,
aC aP
+
--------
pf
dt
Dt
1
dC aP
1
dC
DC
ax ay
(8)
pf ay ax
It is now necessary to expand (8)
The
coordinate
system
chosen
is
coordinates.
natural
into
the
latitude,
convention in the western portion of the northern
In
this
th,
the
northward.
hemisphere.
the longitude, increases to the west and
system, ph,
latitude,
longitude
is
zero
at
the
equator
and
increases
With this choice of coordinates, the differential
transformations are:
dx = - R. cos(th) a(ph)
(9a)
ay = R a(th)
(9b)
Where R, is the radial distance from the center of
It
is
also
explicitly as,
convenient
to
express
the
the
Earth.
coriolis parameter
PAGE 34
f = 2 om sin(th)
(10)
where om is the angular rotation
(9a),
(9b),
of
the
When
earth.
are substituted into ( 8) the result is,
and (10)
DC
rate
dC
dP
1
dC
----------------------------
Dt
dt
2 om sin(th) Rp cos(th) d ( th) d ( ph)
1
- ------------------------
ac
dP
(11)
a(th) d(ph)
2 om sin(th) R,p cos(th)
This can be modified to give,
DC
dC
Dt
dt
1
]
[ ----------------------]
[ 2 om sin(th) Re p cos(th)]
aP
dP
d(ph)
The data
that
patterns
in
is
used
represen t
the
weather
The
data
commo nly
gives
the
a standard pressure surface at the location of the
of
observation.
equation
to
the middle and upper atmosphere is not expressed
in terms of pressure fields.
height
(12)
8(th) a(ph)
d(th)
usually
dC
Thus, it is
using
a
useful
height
for
to
convert
coordinate.
ideal
gases
and
(12)
The
into
an
hydrostatic
small
vertical
approximation,
valid
accelerations,
gives a relationship between a vertical height
coordinate and the pressure coordinate,
PAGE 35
-g p
dP = ---
(13)
dz
r t
where g is the acceleration of gravity, r is the gas
for
air,
and
T is the temperature of the air.
substituted into (12)
DC
dC
Dt
dt
constant
When (13) is
the result is,
g P
[ ---------------------------[ 2 om sin(th) R,,p cos(th)
ac
dz
dC
d(ph) d(th)
The last two terms in (14)
can
]
*
rT
az
(14)
a(th) d(ph)
be
expressed
as
a
Jacobian
where,
ac
aC
a(ph)
d(th)
az
az
d( ph)
a(th)
J(C,z)
(15)
Thus,
g P
- ---------------------------J(C,z)
2 om sin(th) R p cos(th) r T
dt
dC
(16)
If the assumption is made that the atmosphere is an ideal
then
P/(p r T)
equation.
is
unity
and
can
be
eliminated
in
gas,
this
PAGE 36
DC/Dt is the time rate of change of concentration
with
the
flow
of
particles.
To
obtain
equations, DC/Dt is set equal to the rate at
are
redistributed
processes,
such
a
final
which
moving
set of
particles
or removed from the atmosphere by physical
as
horizontal
and
vertical
diffusion,
radioactive decay, and deposition on the ground.
In the model
studied here, the assumption is made that the diffusion can be
modeled
in
terms
of. a
single
diffusion
constant.
The
simplified form of the diffusion is assumed to be,
2
DC
=
--
D V (C)
(17)
E
Dt
where Dis the diffusion constant.
natural
In
coordinates,
this expands to,
1
DC
D
E
Dt
1
a
ac
dBRe
aR,
2
a
cos(th)-----
Deposition
is
radioactive
+------------
d(th)
Re cos(th) a(th)
dealt
decay
with
ac
1
aC
in
(18)
Re sin (th) 6(ph)
a
later
section
and
the
of the particles is neglected in the model
studied here.
The model used in
this
study
contains
only
a
single
PAGE 37
layer.
The
assumptions
essential
in
approximation are that the concentration
the
cloud
debris
and
the
significantl y with height
a
certain
layer
thin
nearly unifo rm winds
assumption
wind
this
of
layer
single
particulates
veloc ities
do
not
vary
or that the par ticles only exis t in
hat
is
ot her
and
greatly simpl ifie
charact erized by unifor m or
physical
Th is
prop erties.
s the computa tional probl em,
but
neglects important vertic al v ariations in the at mosphere.
real
particular,
winds
do
show
In
consid erable shear in both
direction an d speed throu gh t he depth of t he atmosphere.
one
layer
model
always being
used
advected
here
by
assumes
winds
that
The
that the particles are
are
present
one
at
particular l evel.
The first term in (18) contains the radial
concentration.
variation
of
This term can be eliminated because:
Ro= R0 + z
(19a)
Where R. is the radius of the Earth.
The differentiatial form of (19a)
is:
(19b)
dR = dz
Thus,
dC
dRe
dC
-- = 0
dz -
In the single
layer approximation
(19c)
PAGE 38
Thus, by substituting
terms,
the
local
rate
(18)
of
into
(16)
and
rearranging
change of particle concentration
becomes,
-- =D
dt
aC
[
1
a
[--------- -----
dC
E
[
+
cos(th) d(th)
Recos(th) a(th)
t
2
a c
1
Re sin (th)
+--------------
a(ph)
g P
--------------
J(C,z)
(20)
2 om sin(th) Re p cos(th) r T
As will be seen in the trials calculated in
finite
difference
approximations
used
advection introduce numerical diffusion.
is
already
section
to
IV,
the
calculate
the
Since some diffusion
present, and it is desirable to study the model's
properties without physical diffusion, the constant D in
was
set to zero in the trials presented here.
here includes the
appropriate
diffusion
diffusion
term
constant
for
should
detailed trials and in order to make accurate
the
deposition
onto
The derivation
completeness
be
chosen
and
an
for more
predictions
the grou.nd of radioactive particles.
crude calculation of deposition rates is discussed in
IV.3.
(20)
of
A
section
PAGE 39
III. Derivation of finite *difference forms
In order
to
solve
this
equation
numerically,
necessary to convert it to finite difference form.
it
is
The finite
difference analogues to the diffusion terms can be found in
straightforward
manner.
The
standard
two
point
center
differenced expansion of a derivative gives,
I
1
bcW
b(A)
~
C(f
'814I
~
f0A,)
-c(1,)
4 /42
+Of
2j
2p
and,
(4)
(z)
(
I111
ATh
g~2
('h)
4
C
-4
Thus,
*1 £'A)(c(iI.1AiL)-(t~))-~as(
IL-
04) -qtk-
304)(
(23)
The second order derivative in (20) can be found by using
Taylor expansions near the point in question.
I
)
2
z (P
(
(tk)
f
((rL fa p4)
4P4
apj
2
PA) z
a6fh
10[4 (t14)J
(P4) 2
-
((p L-Af4)
{ t,)
two
.*
4
{'
(f4
2
fAr)(f
)
(2')
T he
o
f(On1 of (24)
and
(25)
gives,
PAGE 40
(24
12
thus
-
(Afh)
To
approximate
(27)
ph) - 2 C(k4)
(pl,
the
L
2
Jacobian
advection
by Arakawa (1966) was used.
described
1
term,
the
method
This approximation for
the Jacobian is accurate to high order in spatial
derivatives
and is resistant to computational instabilities that can arise
The Arakawa scheme should
in long term numerical integration.
also
conserve
concentrations
the
in
square
the
of
cloud.
the
area-weighted
particle
The
approximation
for
,
(f k + ,f L
the
Jacobian is,
.z(f,
(CP44#A,)-
(LA 4h))
t,1ft4)-2( , t4ffs))(d(LhA)-C rA'f k],
t- 4tlA)2(4fh, I4A) (p,-4
+ )(C(f, R#LtA1)-(rfth))
-jt
+(p'f puf,ttj (l'Lasfk1I.s ) (fL-ag, - 2t.)
f(i2('f4-Af'4,
-]-z( p-Af l,
-
{
(f4,
-AM)-2( p L.,)
C([Li4)-
((A,-L
fi~df,]
th](
[ (h] -((fJ ,tt-f4]
PAGE 41
The Arakawa scheme was developed in order
advection
Jacobian
towards
equation
for
have
the
conserve certain properties and be stable
to long term numerical integration.
directed
to
the
solution
Arakawa's derivation
of
two-dimensional,
the
was
vorticity advection
incompressible,
flow.
That
equation is:
0
d
--
+
vV
(28.5)
0
dt
is the vorticity, k X V - v.
where v is the velocity and
vorticity,
The quantities tha t were to be conserved were mean
mean
of th e flow, and mean square vorticity.
ene rgy
kinetic
The first constrai nt just fo rces the conservation of the scalar
quantity
that
is undergo ing
advection.
constraints were derived for incompressible, two
flow
by
Fjortoft (1953).
mass,
the
field
concentrations,
dimensional,
require
that
the
total sum of the squares of the masses, and
the kinetic energy of the flow
velocity
two
These constraints, when applied to
the concentration advection equation, (1),
total
latter
The
the
constant.
by
altered
not
is
remain
kineti c
energy
the
constraint
Since
the
advection
of-
is not
of
interest here.
The stability of the
investigated
by
Lilly
second
(1965).
order
Arakawa
scheme
was
He studied certain stability
PAGE 42
measures under several
time differencing
schemes.
With
the
differencing used here, the Eulerian method, (30), Lilly
time
found that the Jacobian could become unstable when applied
(28.5).
It
be
should
to
that the coupling between the
noted
vector and scalar fields that is present in (28.5), is absent
in the concentration advection equation that is used here.
The conservation properties of the scheme were studied by
The
(1970).
Arakawa
Small
almost exactly conserved.
Arakawa and Lamb (1977).
by
conserved
errors
found
were
to
be
this quantity due to time differencing errors,
in
introduced
square vorticity was found to be
mean
the
While the mean vorticity is
exactly
the mean square vorticity is only
scheme,
semi-conserved to within time differencing errors.
scheme
Numerous investigators have used the Arakawa
Brethe rton and Karweit (1975) used the
numerical integration.
fourth order scheme with the same conservation
solve
the
vorticity
used the sec ond order
dimensional
properties
equatio n fo r ocean flows.
scheme
turbulence.
to
study
(1967),
of
an
two
of
simulations
Two det ailed studies of the Arakawa
compared
the
schemes
difference s chemes in their ab ilit y
region
to
Lilly (1969)
scheme and other alternate met hods have been done.
Molenkamp
for
arbitrary
to
sca lar
characterize d by purely cyclon ic f low.
to
The first,
accurately
quantity
finite
other
advect
in
The second and
a
a
field
fourth
order schemes were fou'nd to produces solutions which preserved
PAGE 43
the mean vorticity and the mean square vorti city.
The Arakawa
compared favorably with other simpl e schemes in terms
schemes
of prese rving the peak value of the
quantity
scalar
terms of the velocity of the peak value.
and
in
Negative values were
produced at some gridpoints, but their detailed nature was not
discussed.
Molenkamp's pape r only considered the first forty
timesteps of an advection.
0rzag (1971)
studied
with similar velocit y and scalar fields.
schemes
show that the schemes do pres erve the basic
scalar
field,
patterns.
the
Arakawa
His results
features
of
the
but that ther e are significant.problems in the
Spe ci fically, ther e were a
number
of
what
Orzag
called "wakes of bad numbers" upwind of the main distribution.
These wakes were attributed to phase lag errors in the Arakawa
scheme.
The
wakes
were
said
to
be
components
distribution t ha t were laggin g the main area.
the
informati on
not centered in
wakes
(1971),
is
p.
shown
84.
necessary
the proper
below
in
Thus,
of
the
some
of
to construct the distribution was
diagram of
location.
A
figure
reproduced from Orzag
1,
these
PAGE 44
Figure 1:
Wakes in Orzag's advection trials
The substitution of (23), (27),
and (28) into (20)
gives,
PAGE 45
~((pL7 h)- c
s(A# -^4)(c(M
cos
Cl%
+g -----2 'f
+
~~--- cAk
hi)- ( ,
-14}))
f
PL 4p)(PCh1 4 ,14)-fC(p,,pM)
4t L ohf6,1- ZClfi, i)* c~h
(~L
tk
(a4(sth]
(2 (-A/
iLt-,6- i)
(t
*t A4 ) (4j- t , f'AI
( ., As
))(((f4)(pI
i))
(29)
+(
(
Li( p I
4(p,
14]
fQg,-
Z( L,4f Af,))(4i
-- 2( f-Ap4,t
{ tif1apH )
1
4p,
1t4)A
i))( c(f4, i)- c(7 4Afh
,af -), ](
( 1(4 t,4f4)-2 (
(it-
4 i- Al}
C(f-A ,lf4l )- f f
1))
c14)
14-A14))
I)
f4p t)- 414st f4 -A
This equation is suitable for numerical integration.
PAGE 46
IV. Trials
A. Description of computational method and data
from (29)
In order to calculate dC /dt
it is necessary to
the field of constant pr essu re height.
pressure values are obse rved every
hundred
The heights of certain
hours
twelve
acros s
the
by
North
several
American
The height f ield of consta nt pressure can be found
continent.
by
sca tter ed
stations
know
bet ween
interpolating
all
of
these
stations.
The
interpolation scheme use d in this pape r is not standard and is
discussed in Appendix A.
The
scheme
sufficiently
produced
smooth
fields
that
were
realistic,
and, when contoured, agreed closely with
with iso-height contours that were produced on NMC analysis of
the
same
data.
In the trials tested here, the height field
was artificially constructed to produce pure zonal winds.
interpolation
scheme
was
used, but in the case of idealized
winds, a generating function could have been substituted.
intent,
again,
was
to
The
provide
the
groundwork
for
The
more
sophisticated trials with real wind fields.
The method of computation was quite
51
by
151
straightforward.
A
grid represented the North American region with a
grid spacing of 1/2 degree in
both
latitude
and
longitude.
PAGE 47
The
boundaries of the grid were 30 degrees north latitude and
55 degrees north latitude and 55 degrees
130
degrees
west longitude.
west
longitude
Since no longitudinal variation
was present in the wind field or in the grid spacing that
used,
not
and
was
the particular longitudinal boundaries of the grid were.
important
in
the
advection.
The
computation
was
initialized with a starting concentration field and an initial
height field.
From this initialization, the concentrations at
all points were calculated by a backwards differencing scheme,
also known as the Euler method:
C(t) = C(t- At)+ dC/dt
(30)
This backwards differencing scheme is
The
concentration
height field
was
field
not
was
accurate
to
order At.
updated each timestep, but the
changed.
Upper
atmospheric
weather
systems tend to move more slowly than surface systems and much
more slowly than the wind velocity field
embedded.
Thus,
it was
reasonable
variation of the height field
periods.
over
in
to
which
they
are
the
time
short
time
neglect
sufficiently
By making this approximation, the model essentially
It
moves particles along the streamlines of the flow.
be
simple
to
would
modify the scheme to allow for the updating of
the height field at appropriate intervals
Longitudinal grid spacing varied
southern
boundary
of
the
grid
to
from
48200
m
at
the
31900 m at the northern
PAGE 48
boundary.
This variation is shown below in figure 2.
Figure
Longitudinal gridpoint spacing
Latitudinal grid spacing was constant and equal to
meters.
The
result
of
this
introduce a asymmetry into the
difference
pa rticI e
In fact, a symmetrical' concentration
55700
in spacing was to
advection
pattern
patterns.
in this
grid
PAGE 49
space would indicate an unsymmetric mass distribution
in
the
actual cloud.
Much
of
explosions
the
is
debris
deposited
that
at
is
thrown
levels
high
corresponding to pressures of 10000 pa to
kg m-1 s-2 = 0.01 mb
( 1 Pascal = 1
up
nucl ear
by
in the atmosphere
30000
Pa (Pascals)
(millibars)).
At
these
levels the winds in the jet stream can be as high as 80 m s-1.
In order to keep
criterion
the
the
sta ble
computation
be
to
CFL
number must be less than unity, thus,
Courant
the time necessary for a particle to
must
according
traverse
one
gridpoint
much less.than one time step (Roache, 1976).
With a
time step of one minute this condition was always satisfied in
the
model.
In
the
worst case, with 80 m s-1 winds blowing
zonally across the northern edge of the grid, the time that
particle
take
would
to
cross
one
larger than one, one minute, timestep.
a
gridpoint would be much
This is shown
by
the
calculation below.
grid
spacing
time
-------
gridpoint
v
31900 m
--------
= 399 s = 6.65 timesteps (31)
80 m s-1
This gives a Courant number of 1/6.65 or 0.15.
These trials were run on a Digital Equipment
Corporation
PAGE 50
PDP-11/60
computer.
computer
facility
Physical
in
the
Oceanography
at
Technology.
were
many
inst ances
unsmoothed
of
calculate
was
Department
the
tria ls
when
from
the
other
described
comput ing
time.
part
of
jobs
Institute
fast
couild
on
and
Meteorology
Massachusetts
trials
of the MC IDAS
the
of
there
but
be calculated
system.
The
below required about one to two
The
advection
program
did
not
the concentrations for any gridpoi nts that were far
removed from the cloud.
concentrations
for
limited spatial
range.
adjacent
computer
This computer was not extremely
without competit ion
days
The
or
It was
these
not
necessa ry
to
calculate
grid boxes becaus e (28) has a very
( 28) only affects grid boxes that
are
diagonal ly adjacent to a non-zero conce ntration.
The skipping of many gridpoints al lowed the smoothed trials to
run
much
clouds.
the
due to the smal 1 dimensions of the smoothed
faste r
Integer arithmetic was used throughout
advection.
to
calculate
This arithmetic saved computational time and
space but result ed in truncation errors in some of the
trials
that were much Iarger than would have been present if floating
point arithmetic had been used.
descriptions
of
As will be discussed
in
the
the trials, some of the trials were computed
with integer ari thmetic that used truncation rounding and some
of the trials were computed with nearest integer rounding.
In order to test the properties of the
in
the
advection
scheme,
the
sum
of
Arakawa
the
Jacobian
area-weighted
concentrations and the sum of the squares of the area-weighted
PAGE 51
concentrations
was
The
calculated.
conserved if the scheme properly
first
conserved
quantity
mass.
was
This
was
because the area weighted concentration in a grid box was just
the number of particles in that grid box.
conserved
exactly
by
This
quantity
is
the Arakawa scheme, even in its finite
Variations in this quantity are a meas'ure of
difference form.
arithmetic truncation error due to the integer arithmetic used
The Arakawa
in the trials studied here.
conserve the second quantity.
This
is
not
should
also
This second quantity represents
the resistance to spurious diffusion
scheme.
scheme
caused
by
the
numeric
conserved exactly and is a measure of
time differencing errors in the finite difference scheme.
To simp 1ify much of the ear ly testi ng a simpl ified heig
field
was
used
The
height
longitudinal var iation and thus,
a
pure
wes terly
figure 3.
wind.
field
by (6a)
that
was
and (6b)
used had
represented
This height field is shown below in
PAGE 52
s-s.
e
So
'4~-e.
4..
5
Va0
4-
-4
-x
4-
IrI
97120
F0igre
Figure
Height
.
field used in the trials
The heights dropped uniformly
of
(10)
latitude.
gives
-
The
combination
240 m every
10
degrees
of equations (6b) , (9b) , and
PAGE 53
1
U
=
-
dP
-
(32)
-
----------------
-
2 om sin(th) Rep
d(th)
which, assuming an ideal gas, reduces to,
dz
u
=
-
(33)
--------------
2 om Re sin(th)
d(th)
This gave a wind speed of u = 14.54/sin(th) m s-1 in the
simplified field.
[
[
U=- [
In grid units the wind was,
I
-- -g
-- - - - - -
2 om Rrsin(th)
I
*
az ]
1 gp
2 pi Recos(th)(0.5 degrees/360 degrees)]
Where gp stands for gridpoint .
for
the
physical
constants
][ a(th)]
By substituting actual
(34)
values
in the above equation, the wind
velocity was found to be:
-4
2.613 x 10
U
=----------------
gp
s-1
sin(th) cos(th)
where tstp indicates timestep.
1.568 x 10
--------------sin(th) cos(th)
gp tstp-1 (35)
PAGE 54
The
wind
coordinates,
study.
not
independent
of
latitude
in
north
6.03 x 10~
longitude,
of 5.22 x 10
gp s-1 and 5.56 x 10
northern
these
but. the variation was small in the domain under
In the grid units, the wind reached a minimum,
degrees
and
was
boundaries
of
gp s-1
at
45
gp s-1 and rose to
at
the
the grid, respectively.
southern
In the
trials the cloud was initialized at the center latitude of the
grid,
at
latitude
42.5
degrees
north.
In this region the
latitudinal shear of the wind was only of the order of
percent
over several degrees.
a
few
The longitudinal shear was, of
course zero since there was only zonal wind in
these
trials.
The latitudinal variation of the wind is shown below in figure
4 and figure 5.
PAGE 55
1.00
24.00
VELOCITY (M/S)
Figure 4:
Wind speed versus latitude in m s-1
PAGE 56
0.61
Figure
Wind speed versus latitude in gridpoints s-1
Using this simplified wind field a number of trials were run.
PAGE 57
B. Individual trials
1. Unsmoothed trials
a. One point i nitialization
i.
Description of initialization
In the first test case, a concentration of
was
placed
in
a
grid box
longitude of 92.5 degrees.
in
the
model
10,000
units
a latitude of 42.5 degrees and
The location of the initialization
was at gridpoint (76,26) with the longitudinal
gridpoint listed first, as will be
the convention
throughout
this paper.
ii.
Diagrams of the advection
The resulting model advection
through 6g.
6a.
is
shown
in
o
0
0
0
0 100
0
0
0
0
One point initialization unsmoothed:
advection pattern at t=0000
0
0
0
0
0
-26
-105 10000
0
-26
0
0
0
0
6b.
0
0
26
105
26
0
0
One point initialization unsmoothed:
advection pattern at t=0001
figures
6a
PAGE 58
25.-
6c.
0
0
0
0
0
0
0
0
0
0
0
-1
-52
-210 9998
-1
-52
0
0
0
0
0
0
52
21L0
0
0
One point initialization unsmoothed:
advection pattern at t=0002
25-
6d.
0
0
0
0
0
0
0
0
0
0
0
-789
-315 9993
-789
0
0
0
0
-3
-3
0
0
79
315
78
0
0
0
0
0
0
0
0
0
One point initialization unsmoothed:
advection pattern at t=0003
2g -
0
0
0
0
0
0
0
0
6e.
0
0
0
-258
-1044
-259
0
0
0
0
-6
-49
-6
0
0
0
0
c2t8
1C>44
259
0
0
One point initialization unsmoothed:
advection pattern at t=0010
0
0
0
0
0
0
25-
0
0
0
0
0
0
0
0
-38
-113
-164
-116
0
0
0
0
0
0
0
17
0
-94
89
359 -911
590 -1624
363 -816
-95
90
18
0
0
0
0
0
6f.
0
0
0
0
0
0
-24
-43
78
-166
511
107
441 -2759
986
3439 -4739
1887
990
449 -2765
-169
512
109
-44
-26
78
0
0
0
0
0
0
0
0
24
-511
-441
4738
-449
-512
26
0
0
0
0
-43
-166
996
3439
990
-169
-44
0
0
0
0
-17
94
811
1624
816
95
-18
0
0
0
0
0
89
359
580
363
90
0
0
0
One point initialization unsmoothed:
advection pattern at t=0100
0
0
0
38
113
164
116
39
0
0
0
-J.
mLIi
,00MRooS
WOO
-
wotoOQ
emrj MOAT. . .
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PAGE 60
iii. Description of advection
side
A problem developed immediately at the lee
The
cloud.
appeared in timestep 0001.
This
south
of
gridpoints
In timestep 0002, a
negative
the
behind
the
include
the~i nitial burst.
positive wak e had developed
first
wake
By timestep 0002 the negative wake
had grown larger and had exp anded to
and
the
model produced a wake of negative concentrations
upwind of the original burst of particles.
north
of
The
one.
positive and negative wakes continued to grow in magnitude and
at timestep 0010,
behind
the
another
trailing
negative
posit ive one.
wake
began
In subsequent timesteps,
the wakes continued to grow in a similar fashion.
0020
the
negative
first
wake
had
wrapped
northeast and southeast of the main cloud.
first
positive
wake
had
begun
develop
to
to
wrap
In
In timestep
to the
around
addition,
the
the first
around
negative wake.
The wakes are
timestep
0100
a
serious
problem
in
this
model.
they had begun to dominate the advection.
gridpoints with non-zero concentrations were counted;
By
The
54 were
in the wake while only 25 were in what appeared to be the real
cloud.
By timestep 1440, one day after the initial burst, the
grid was covered by a confusing field of positive and negative
concentrations with a small positive cloud at its front.
peak
negative
The
value tended to be found in the first negative
wake to the southwest 'of the positive area.
A table comparing
PAGE 61
the peak values in the positive area to the peak values in the
first lee wake is shown below in table 1.
timestep
largest
positive value
ratio
maximum first
wake value
0300
1749
0900
784
-666 (maximum value in
any wake was a +894)
-1.17
(.88)
1200
657
-624 (maximum value in
any wake was a +760)
-1.05
(.86)
1440
614
-573 (maximum value in
any wake was a -631)
-1.07
(-.97)
Table 1:
-1821
-. 96
One point initialization unsmoothed:
magnitude of wake and the forward positive area
With this single gridpoint initialization, the absolute
value
of the concentrations remained perfectly symmetrical about the
The only difference between the upwind and
starting latitude.
downwind
concentrations
was
the
region
of purely positive
concentration at the front of the concentration
The
cloud
east-west axis.
was
not
perfectly
symmetrical
lost.
about
an
Although in the early timesteps the cloud was
nearly symmetrical about this axis, much of the
eventually
distribution.
symmetry
was
This was not unexpected since the gridpoint
area and the wind were not latitude independent.
These
wakes
appear quite similar, qualitatively, to the wakes found in the
PAGE 62
advection study by Orzag (1971) and illustrated in figure
iv. Cross-sections
Cross sections of the concentrations along a latitudinal
through
the
initialization
latitude ( 42.5 degrees north or
grid level (x,26) ) are shown below in figures 7a
for
selected
timesteps.
line
These
show
the
through
process
development along the axis of movement of the cloud.
7g
of wake
PAGE 63
7a.
One point initialization unsmoothed:
cross-section at t=0020
PAGE 64
66.00
76.00
LONGITUDE GRID#
7b.
One point initial ization unsmoothed:
cross-section at t=0050
PAGE 65
66.00
76.00
LONGITUDE GRID#
7c.
One point initialization unsmoothed:
cross-section at t=0100
116.00
126.0E
PAGE 66
126.00
7d.
One point initialization unsmoothed:
cross-section at t=0200
PAGE 67
LONGITUDE GRID#
7e.
One point initialization unsmoothed:
cross-section at t=0300
PAGE 68
CONCENTRATION X-SECTION OF TIMESTEP 0600
B~
X
t"
0
II
*
E
cc~'
j
Z
V jV
24
0
LI
*
*1
I
I
A
I
I
_________
__________
_____
____A
I
I
I
____
_____
____
_
26.00
___ ____
_________
________
I
I
I
I
A
I
I_____
I
1
I
I
______________________________________________
______________________
______________________
______________________
______________________
______________________
_____________________
____________________________
____________________________
+
I
46.00
56.00
66.00
76.00
86.00
96.00
106.00
116.00
1Z6.00
I
I
LONGITUDE GRID#
7f.
One point
initialization unsmoothed:
cross-sect
ion
at t=0600
PAGE 69
ICONCENTRATION
ABWI
X-SECTION OF TIMESTEP 1440
co
i
Z:
0
CE
cc:
_
_
_
1
J1
_
_
_
_
1
:
-.
0
Li
I_____
_____
k
____
__________________
_____
3
___________
F1
26.00
36.00
46.00
56.00
66.00
76.00
86.00
96.00
106.00
LONGITUDE GRID#
7g.
One point initialization unsmoothed:
cross-section at t=1440
116.00
PAGE 70
The graphs
show
magnitude,
of
number
and
in
Also shown is the decrease in the
area
forward
the
in
magnitude
peak
wakes.
the
in
growth
continued
the
caused
numer ical
by
diffusion in the model.
compared
In early timesteps, the wake was small
peak
positive
As
from
its
initial
in
The wakes in.the early timesteps we re
longitudinal
the
But,
direction.
varying
rapidly
by timestep 0300, an
The clou d began to develop a
interesting phenomenon occurred.
wake to the immediate rear of the f orward positive area
large
that had a great longitbdinal extent.
and
comparable
larger than the peak values in the forward positive
even
area
grid
soon as the cloud began to drift significantly
from its starting location, the wake value s became
and
the
But this was only because the
concentration.
peak concentration had not yet moved
loca tion.
to
by
timestep
1440,
a
of long period smooth wakes
train
separated
and
trailed the forward area
Thi s process continued,
rapid
the
it from
oscillation s that were still
present at th e rear of the cloud.
The process responsible for
this
probably
a
reduction
diffusion i nherent to
that
of
the
the
smooth
wake
shedding
was
gradients in the cloud by the
finite
differ ence
were made in the numerical scheme.
approximations
The model apparently
a
long
period of t ime and thus slowed the wake fo rming process.
This
reduced the gradients around the positive
may have made the pattern look mor
reduce
the
problem
introduced
area
over
pleasi ng, but it
the wakes.
did
not
In the later
PAGE 71
smooth
timesteps,
the
comparable
in magnitude to the forward area.
boundaries of the "real"
became
wakes
mostly
positive
and
Thus the actual
cloud were difficult to ascertain.
Mass in the forward positive area
v.
It was tempting to try to interpret the forward
as
area
the
actual
cloud
and the wakes as just a spurious
byproduct of the Arakawa scheme.
not
was
justified.
However,
this interpretation
The number of particles in the positive
area grew quickly to significantly
initial burst.
positive
more
particles
than
the
The table below shows the sum of the values of
the concentrations in the positive area grid boxes at selected
timesteps.
timestep
sum of concentrations
0
10000
1
10288
2
10315
100
17527
1000
15918
1440
14532
Table 2:
percentage over t=0000
One point initialization unsmoothed:
excess concentration in the forward positive area
PAGE 72
The
of
mass
the
including
the
concentrations,
of
was- conserved to within a few percent
wakes,
(see section vii).
distribution
entire
This indicates that the mass in the wakes,
in the later timesteps, while nearly adding to zero, must have
mass
had a negative component representing about one half the
of the cloud.
The
sum
of
weighted
and
thus
is
not
above
shown
area
not
extent,
north-south
With
of concentrations should be closely conserved.
sum
mass
perfect
concentrations
conservation,
the
in
change
the
concentrations
would
would increase
but
The
percent.
eight
drop because the area-of the grid boxes
the
number
of
particles
remain
would
The number of particles did not remain constant if
constant.
the positive area was interpreted as the "real"
It is notable that, despite the
thousands
of
sum
in a cloud that moved from 45 degrees north to
40 degrees north latitude only be minus
of
grid
points
filled
that
fact
with
This
was
cloud.
were
there
wake particles, the
alternating positive and negative wakes did,
zero.
is
conservation of mass
a
exactly
However, in a cloud of small
indicator.
the
concentrations
almost,
add
to
evident because the model conserves mass to
within 20 percent (see
discussion
below)
and
the
positive
region's sum of concentrations did not differ -drastically from
the initialization value of 10000 units of concentration.
PAGE 73
vi. Velocity of forward positive area
Another difficulty with interpreting the positive area as
the
latitude
the
its
was
cloud
actual
winds
been located at gridpoint
inspection,
it
was
At 42.5 degrees north
at
particles
carry
should
gridpoints per timestep.
velocity.
0.03147
At this speed, the cloud should have
(79.1,26)
obvious
by
timestep
0100.
the bulk of the region of
that
positive concentration was far upwind of gridpoint (79,26)
that
time.
By
timestep
gridpoint (107.5,26).
By
at
1000 the cloud should have been at
It was clear again that the
region
of
positive concentration lagged behind this location.
The wakes did have signif icance in the advection speed of
the
cloud.
If
the
Model advected the cloud at exactly the
wind speed, then the centroid of the cloud would
been
at
gridpoint
(76+.0314 7*T,26)( see figure 5 ).
wakes were included in a calcu lation of the
have
always
centroid
If the
of
the
cloud, then the centroid did move at the predicted velocity of
the cloud.
For example, at timestep 0001 the centroid of
positive area was at gridpoint (76.0155,26).
of the entire cloud,
including the
wake,
predicted gridpoint, at (76.03 14,26).
was
the
But the centroid
close
to
the
PAGE 74
vii. Conservation of mass
The Arakawa scheme should conserve both the area weighted
sum
of
the
concentrations
at
all
gridpoints
and
area-weighted sum of the squares of the concentrations at
gridpoints.
cause
all
The first is just a conservation of mass criteria
and the model did
likely
the.
of
fairly
the
was
the
from perfect conservation.
The
well.
drift
Truncation
model lost about one fifth of the particles by
but began to regain them by timestep 1600.
error
timestep
1200
Though the loss of
particles appears systematic, it is possible
that
first
the
1600 timesteps are just representative of the first portion of
a random walk about pure conservation.
program
that
version
of
the
performed this particular trial was lost and it
is not known whether it used truncation
integer
The
rounding
in
rounding
its integer arithmetic.
or
nearest
A graph of the
time history of the mass of the cloud is shown below in figure
8.
PAGE 75
60.00
80.00
100.00
TIME (MINUTES)
Figure 8:
(X10
One point initialization unsmoothed:
normalized mass
PAGE 76
viii. Conservation of squared mass
The sum of the area weighted
presented
more of a problem.
squares
of
concentrations
This quantity grew smoothly and
systematically to a value that was 69 percent higher than
initialization
value.
This
monotonic
growth
explained by arithmetic truncation error and
rooted
in
spurious
numerical
diffusion.
its
The
cannot
origin
Very
roughly,
diffusion
values
are
produced
by
of totally positive
sum of the masses is conserved.
timestep.
squares
it
while
While the increase of 69
percent that occurred over the first day of advection was
desirable,
When
diffusion into the wakes,
however, the sum of the squares of the masses may go up
the
was
section
quantities would result in a decrease in this quantity.
negative
be
mechanism
responsible for this growth is discussed in detail in
IV.B.2.a.vii.
its
not
represents an increase of only .05 percent per
A graph showing the time history of the sum of
the
of the area weighted concentrations is shown below in
figure 9.
PAGE 77
NORMALIZED SUM OF CONCENTRATIONS
to
I
-I
00
:
20.00
40.00
I
iI
60.00
80.00
100.00
TIME (MINUTES)
Figure 9:
ABW
120.00
(X10 1 )
One point initialization:
normalized mass squared
I
140.00
160.00
180.00
PAGE 78
ix. Discussion
The wakes make
calculating
this
advection
model
clearly
unsatisfactory
with the one gridpoint initialization.
This failure stems from the errors introduced
gradients
of
for
concentration
initialized gridpoint.
that
Since the model
were
by
the
present
produced
strong
at
the
alternating
wakes, the gradients were only slowly
positive
and
reduced.
Thus, the wakes continued to grow to the rear of the
cloud.
only
The
large number
numerical
negative
had
iterations
of
in wake frequency arose after a
reduction
diffusion.
reduced
the
gradients
by
Even this did not reduce the amplitude
the only reduction was in their frequency.
of the wakes;
b. Nine point initialization
i. Description of initialization
In order to reduce the wake by reducing the gradients
of
concentration, a multi-point initialization was attempted.
In
this initialization, a three by
initialized
gridpoint.
north
with
units
of
gridpoint
region
concentration
in
was
each
The center of the area was, again, at 42.5 degrees
latitude
(76,26).
1111
three
and
92.5
west
longitude,
or
at gridpoint
PAGE 79
ii.
Diagrams of advection
The results of the advection are shown below
10a through lOf.
2 S
-
0
0
111.1
1111
111:1
111.1
1.111
1111
0
1111
111:1.
111.1.
0
0
0
0
0
0
0
10a. Nine point initialization unsmoothed:
advection pattern at t=0000
0
0
0
0
-
0
0
0
0
-3
-15
-17
-15
-3
0
0
-3
1096
1094
1095
-3
0
0
0
1111
1111
1111
0
0
0
3
1126
1122
3
0
10b. Nine point initialization unsmoothed:
advection pattern at t=0001
24-
0
0
0
0
0
0
0
0
-6
-29
-34
-29
-6
0
0
-6
1031
1076
1031
-6
0
0
0
1111
1110
1111
0
0
0
6
1140
11.45
1140
6
0
10c. Nine point initialization unsmoothed:
advection pattern at t=0002
in figures
PAGE 80
25 ..
0
-9
-43
-51
-43
-9
0
0
0
0
0
0
0
0
0
-9
1066
10581
1066 1
-91
0
0
0
11
109
110
0
0
9
1154
1162
11E4
0
4.
4:
e.
10d. Nine point initialization unsmoothed:
advection pattern at t=0003
S7
0
0
-26
-135
-162
-136
-26
0
0
0
0
0
25.
0
0
0
0
0
0!
0
-30
-6
9571 1093
927 1036
956 1093
-30
-6
0
0
0
0
0
0
26
1246
1272
1245
26
0
0
0
0
30
155
185
30
0
0
10e. Nine point initialization unsmoothed:
advection pattern at t=0010
.7.f
0
0
0
-6
-24
-30
2f- -24
-6
0
0
0
0
0
0
37
83
110
86
37
0
0
0
0
0
0
-66
-191
-259
-198
-66
0
0
0
0
0
-10
37
312
385
312
35
-12
0
0
0
0
0
0
0
0
17
41
26
-98 -286
95
-83
-32
-224
-224 -404
-190
-90
-35
-223
-97 -289
96
186
27
43
0
0
0
0
0
0
0
0
-565
-301
812
624
606
-305
-56
0
0
0
0
-64
83
1174
1388
1173
61
-64
0
0
0
0
-8
245
669
1106
872
246
-9g
0
0
0
0
0
173
433
552
436
175
0
0
0
10f. Nine point initialization unsmoothed:
advection pattern at t=0100
0
0
0
71
160
201
162
73
0
0
0
PAGE 81
iii. Description of advection
The wakes, as expected, developed more slowly.
negative
wake
developed
immediately,
but
timestep 0003 until the first positive wake
at
timestep
concentration
serious
0100
At
However,
the
positive
was
filled
developed.
Even
area.
with
were
still
a
timestep 0100 the number of grid boxes
This
occupied
by
the
was better than the one point
start, but was still unsatisfactory.
grid
until
wakes
occupied with wake was 51, compared with 31
leading
it took
the initial cloud area still dominated the
pattern.
problem.
The first
By
timestep
1000
the
a huge array of positive and negative
wakes trailing a small positive region.
The wakes near the-front of the cloud
smoothly
varying
oscillations.
highest
axis
became
and of a longer period than the rapid early
The ratio of the
highest
wake
value
to
the
forward peak value is shown to decrease to near unity
in the table below.
the
eventually
chosen
The highest wake value was not
found
on
for the cross sectional graphs and thus the
table indicates, perhaps more
realistically
sections, the scale of the wake problem.
than
the
cross
PAGE 82
timestep
largest forward
positive value
ma ximum
wake value
ratio
N.A.
0000
1111
no wake
0001
1128
-17
-66.4
0002
1145
-34
-33.7
0010
1272
-162
-7.9
0050
1483
-464
-3.2
0100
1388
-404
-3.4
0300
1075
-628
-1.7
0600
772
-489
-1.6
0900
012
-430
-1.4
1200
561
-421
-1.3
1440
533
-400
-1.3
Table 3:
Nine point'initialization unsmoothed:
magni tudes of wake and the forward positive area
The
symmetry
initialization
that
was
not
was
evident
here.
windward side of the cloud tended to
The
value.
largest
values
in
present
of
be
the
one
The
valu es
larger
in
point
in the
absolute
concentrations (positive or
negative) were always found in the forward positive area,
were,
of
course,
positive.
and
There was not a mirro r image of
high alternating positive and negative values at the rear.
PAGE 83
iv. Cross-sections
Cross sections through the cloud
at
are shown below in figures 11a through 11f.
selected
timesteps.
PAGE 84
66.00
76.00
LONGITUDE GRID#
11a. Nine point initialization unsmoothed:
cross-section at t=0050
00
126.0
PAGE
CONCENTRATION X-SECTION OF TIHESTEP 01g0
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66.00
76.00
LONGITUDE GRID#
86.00
96.00
__
106.00
Nine point initialization unsmoothed:
cross-section
I
at t=0100
116.00
126.0k
PAGE 86
LONGITUDE GRID#
11c. Nine point initial ization unsmoothed:
cross-section at t=0200
PAGE 87
LONGITUDE GRID#
11d. Nine point initialization unsmoothed:
cross-section at t=0300
PAGE 88
66.00
76.00
LONGITUDE GRID#
11e. Nine point initialization unsmoothed:
cross-section at t=0600
PAGE 89
11f. Nine point initialization unsmoothed:
cross-section at t=1440
PAGE 90
These cross sections indicate better
indicated
in
than
results
the one point initialization.
were
The forward peak
was the dominant feature throughout the first day of advection.
Towards
the
end
of one day, however, the wakes were growing
to
the
forward
relative
initialization,
the
peak.
model
As
the
with
to
seemed
point
one
reduced
have
the
destablizing gradients, somewhat, by the later timesteps.
v. Mass of forward positive area
With
slightly
particles.
this
more
initialization,
like
what
was
the
area
positive
became
expected of a "real" cloud of
As discussed above, it was the dominant feature in
terms of the magnitude of its concentrations.
still problems with this interpretation.
But, there were
As in
the
case, the region contained too much concentration.
previous
A chart of
the sum of the concentrations is shown below in table 4.
PAGE 91
timestep
sum of concentrations
0
10000
1
10058
2
10114
10
10520
100
12111
500
14181
1000
14391
1440
14155
Table 4:
percent over t=0000
Nine point initialization unsmoothed:
excess concentration in the forward positive area
The magnitude of the excess was a bit smaller
of
that
the one point initialization, but it was still very large.
It should be noted again that the sum
was
than
not
area
weighted
of
the
concentrations
and is thus not a perfect measure of
conservation of mass.
vi. Velocity of the forward positive area
The velocity of the positive region was
expected
of
an
actual
cloud.
region at timestep 0001 was at
was
closer
closer
to
that
The centroid of the positive
gr idpoint
(76.0202,26)
to the desired gridpo int, (76.03147,26)
which
than than
PAGE 92
in the one point initialization.
The location of the
h ighest
concentrations at timestep 0100 was at gridpoint (78,26) which
was near the predicted gridpoint (79.1,26),
1000
the
means
and
at
timestep
peak concentration was at gridpoint (104,26).
that
predicted
the
positive
position
by
region
about
only
ten
lagged
percent
behind
in
This
the
later
the
timesteps.
vii. Conservation of mass
The area-weighted sum of the concentrations was much more
closely
conserved
in
only about 5 percent.
preferentially
forth
around
lose
about
The maximum deviation was
The
in
mc del,
particles ,
approximate
integer rounding was
trial.
this trial.
used
case,
of- mass.
conservation.
not
Nearest
the integer arithmetic in this
The variation here was consistent with a
perfect
did
but seemed to waver back and
con servation
in
this
random
walk
The time history of the sum of
the area weighted concentration s is shown below in figure 12.
PAGE 93
60.00
80.00
100.00
TIME (MINUTES)
Figure 12:
(X10
Nine point initialization unsmoothed:
normal ized mass
PAGE 94
viii. Conservation of squared mass
As with the one point initialization, the
sum
As
of
area
weighted
the squares of the concentrations grew monotonically.
stated
discussed
before,
in
an
explanation
section IV.B.2.a.viii.
for
this
increase
The time history of the
area weighted squares of the concentrations is shown below
figure 13.
is
in
PAGE 95
TIME (MINUTES)
Figure 13:
(X10
1
)
Nine point initializion unsmoothed:
normalized squared mass
PAGE 96
c. Sixty-nine point initialization
i.
Description of initialization
The results of the
point
nine
initialization
improvement
over the one point
to further
duce the.sharp gra dien ts
and
duce the wake, a m uch smo
her
consisted
of
thus
used.
initialization
This
illustrated
initialization.
was
centered
on
a
evenly in all directions
was
defined
to
spacing.
nit ialization was
69
point
nor mal
distribution
The variance
of
the
distribution
be the distance between adj ace nt gridpoints.
Thus,
the
an d
w as
concentration
latitudinal
symmetric
gridpoint units but the actual mass distribut ion was
skewed.
The
cloud
co ntained more mass at its
than in the north due to the greater
south.
box
grid
in
slightly
southern edge
area
in
the
addition, t he cloud was longer 1ati tudinally than
In
longitudinally due to the greater
All
grid
gridpoin t a nd dropped off
certain
No distinction was made- between longitudinal
grid
In an attempt
The co nce ntrations were
determined by calculatin g the value of a
that
an
the edge of the cloud
timeste p 0000 below.
as
were
calculated
values
were
latitudina 1
grid
spacing.
multiplied by 300 00 in order to
produce integers large e nough so that they would be relatively
resistant
to
truncation
contained 11,968 units of
The
error.
Thus, the central gridpoint
concentration
gridpoints immedi'ately to the north
or
30000*normal(0).
south,
east and west
PAGE 97
of
this
center
concentration
immediately
had
which
to
a
at
contained
440 3
30000*normal(1. 414
northeast,
unit s
V2
=
larger
trials.
by
units
and
southeast,
of V2 gridpoints from the cen ter,
of
concentration
which
is
All boxes with calculated values
).
were
of
gridpo ints
The
northwest,
very
small.
concentration in thi s c loud was 75196,
seemed
7259
ini tialized so that the gradients
cloud
the
of
30000*normal(1).
d istance
above .5 units were
of
was
the
southwest,
edge
concentrations
abs olute
at
the
The total amoun t of
so errors in this t rial
standards
than
in the prev ious
The cloud c ent er was initialized at the same loca tion
as the previously described trials.
In this trial the
cloud
began
advecting
when
it
was
already quite large, covering most of a nine by nine gridpoint
box.
In
real
north-south
units
this
dimension
of
indicated
5 01000
a
meters
cloud
and
that
an
had
a
east-west
dimension of 369000 meters.
ii. Diagrams of advection
The results of the
through 14f.
advection
are shown below in figures 14a
PAGE 98
25
0
0
0
25
-
0
0
0
0
0
0
0
0
0
0
0
1
2
4
2
1
0
0
0
0
0
0
0
0
0
1
18
81
133
81
18
18 219 982 1620 982 219
81 982 4403 7259 4403 982
133 1620 725911968 7259 1620
81 982 4403 7259 4403 982
18 219 982 1620 982 219
1
18
81
133
81
18
0
1
2
4
2
1
0
0
0
0
0
0
1
18
81
133
81
18
1
0
0
0
1
0
2
4
2
1
0
0
0
0
1
2
4
2
1
0
0
0
0
0
0
0
0
0 0
0
0
0
0
14a. Sixty-nine point initialization unsmoothed:
advection pattern at t=0000
75
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
2
1
1
0
0
0
0
0
0
0
0
1
2
4
0
15
76
133
13 197 951 1620
66 916 4307 7259
111 1523 711811968
66 915 4306 7259
13 197 950 1620
0
15
76
133
0
1
2
4
0
0
0
0
0
0
2
1
86
21
1013 241
4499 1048
7400 1717
4500 1049
1014 241
86
21
2
1
0
0
0
0
2
23
96
155
96
23
2
0
0
0
0
0
1
3
6
3
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
14b. Sixty-nine point initialization unsmoothed:
advection pattern at t=0001
2y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
1
2
4
71
133
-1
12
9 176 919 1619
52 851 4209 7256
90 1427 697511964
52 850 4208 7256
8 176 918 1619
-1
12
71
133
0
1
2
4
0
0
0
0
0
0
2
1
91
24
1044 263
4594 1116
7540 1816
4596 1117
1045 263
91
25
2
1
0
0
0
0
3
28
112
178
112
28
3
0
0
0
0
0
1
4
8
4
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
14c. Sixty-nine point initialization unsmoothed:
advection pattern at t=0002
PAGE 99
0
0
0
0
0
0
25- '0
0
0
0
0
0
0
0
0
0
0
0
1
2
4
2
1
-2
9
66 132
96
28
S 166 887 1617 1075 286
39 787 4111 7250 4688 1186
71 1333 683111966 7679 1917
39 786 4109 7250 4691 1186
4 155 886 1617 1076 286
-2
9
66 132
96
29
0
1
2
4
2
1
0
0
0
0
0
0
0
0
4
34
129
203
129
34
4
0
0
14d. Sixty-nine point initialization unsmoothed:
advection pattern at t=0003
7Y.
0
0
0
0
0
0
25
0
0
0
0
0
0
0
-2
-14
-26
-30
-26
-15
-3
0
0
0
0
0
0
0
1
2
4
2
1
-10
31
121 125
56
26 653 1567 1274 457
377 3401 7128 5314 1696
726 5795 11788 8601 2664
376 3396 7126 5318 1702
26 651 1566 1276 458
-10
31
121 125
57
1
2
4
2
1
0
0
0
0
0
0
0
13
86
276
416
277
86
13
0
0
14e. Sixty-nine point initialization unsmoothed:
advection pattern at t=0010
0
0
0
0
0
3
- 0
0
0
0
0
0
0
0
-18
-31
-41
-31
-19
0
0
0
0
0
0
0
0
0
0
0
0
0
44
0 -33
91 -16 -103
38 284 -50 -503 -342
0
23 -79
634
46 -19 -204 449 236 -1262-1167
989 -765 -2357 1023 5367
94 -113 -183
116 -178 -120 1179-1525-2158 3843 9219
992 -754-2378 9865 5346
93 -113 -186
45 -14 -210 451 252-1254-1205 610
35 290 -40 -510 -359
0
23 -80
0 -34
44
0
94 -12 -104
0
0
0
0
0
0
2
0
0
0
0
0
0
-66
123
1868
6238
9181
6250
1866
116
-73
0
0
-13
345
1747
4390
5971
4414
1762
346
-9
0
0
0
19
22
281 153
1048 467
2242 902
2898 1122
2261 909
1062 476
284 157
29
28
0
0
0
4
58
166
297
359
298
169
61
8
0
14f. Sixty-nine point initialization unsmoothed:
advection pattern at t=0100
PAGE 100
iii. Description of advection
The first thing that was immediately obvious was that the
first
negative
wake
previous trials .
In
much more slowly than in the
In timestep 0001 there was no negative wake.
0002
timestep
developed
t here
were
two grid points with negative
concentrations.
Thes e wake values were both much smaller than
with
poi
the
nine
mar ked
especially
concentration.
two
grid
boxe s
concentration.
nt
initialization,
s ince
this
trial
this
had
difference
a. larger
was
total
In ti mestep 0002, at the lee end of the cloud,
con tained
a
single
unit
of
positive
Thes e boxes were relatively isolated from any
large concentrations and were, as a result, trapped.
of
significant
gra dients
occur at all
in this model.
concentration,
Without
no advecti on could
The terms in (29) round
to
in the integer a rithmetic if large
gradients are absent.
result here was that cloud started
to
leave
the
two
zero
The
boxes
and the first negat ive wake actually developed between
behind
the isolated box es and the remaining bulk of the cloud.
was
seen
first in timeste p 0003.
isolated by time step 0005.
timestep
00 15.
It
was
The two boxes were totally
The firs t positive wake formed
composed
of
both
and
the
by
a positive wake
produced by the large negative conce ntrations to its
side,
This
two isolated boxes just discussed.
windward
From this
time on, the adv ection followed the same general pattern as in
the
other
trials.
By
timestep
1440, the grid was, again,
PAGE 101
filled with a confusing
array
of
alternating
and
positive
negative wakes that extended to the rear and wrapped around to
the front of the location
of
the
cloud's
positive
forward
area.
The decrease -inthe ratio of the largest forward positive
to
value
the maximum wake value was slower than the decrease
with the other initializations.
This is shown
in
the
table
below.
timestep
largest forward
positive value
maximum
wake value
ratio
0000
11968
0001
11968
0002
11964
-1
-11964.0
0010
11788
-30
-392.9
0050
11303
-526
-21.5
0120
9219
-1525
-6.0
0300
7243
-2976
-2.4
0600
5365
-3206
-1.7
0900
4364
-2789
-1.6
1200
3747
-2488
-1.5
1440
3421
-2394
-1.4
Table 5:
no wake
N.A.
infinite
Sixty-nine point init ialization unsmoothed:
magnitude of wake and forward positive area
The dis tr-ibut ion of
concentrations
was
not
symmetric.
PAGE 102
The
positive
cloud
dominated the concentration pattern.
In
the early timesteps the positive area concentrations were much
larger
than
the
largest wake values, but, the wakes grew in
relative magnitude as tle advection continued.
The relative number of grid boxes in the wake
slowly,
but
was
still
unacceptable.
boxes were in the wake and 48 were
area.
in
grow
more
In timestep 0120,
the
forward
40
positive
This small difference in favor of the positive area was
erased soon after this timestep as the wakes continued to grow
iv. Cross-sections
Cross sections through the cloud are shown below in figures
15a through 15e.
PAGE 103
15a. Sixty-nine point initialization unsmoothed:
cross-section at t=0050
PAGE 104
15b. Sixty-nine point initialization unsmoothed:
cross-section at t=0180
PAGE 105
66.00
76.00
86.00
96.
LONGITUDE GRIO#
15c. Sixty-nine point initialization unsmoothed:
cross-section at t=0300
PAGE 106
66.00
76.00
LONGITUDE GRID#
15d. Sixty-nine point initialization unsmoothed:
cross-section at t=0600
PAGE 107
15e. Sixty-nine point initialization unsmoothed:
cross-section at t=1440
PAGE 108
The cross sections show the well ordered pattern of wakes
developed, more slowly, behind the cloud.
that
of
the
large
gradients
suppressed the
rapidly
evident in the
other
in
the
cloud
The reduction
had
successfully
oscillating, chaotic pattern that was
initializations.
The
varying
slowly
oscillation was still present and, by timestep 1440
began
be comparable in amplitude to the forward positive area.
v. Mass of the forward positive area
The problems in interpreting
the
forward
as
area
the
"real" cloud that were present in the previous initial izations
were
still
present
concentration
in this
trial.
The
sum
of
the
in the forward area is shown in the table below
for selected timesteps.
timestep
sum of concentrations
% over timestep 0000
75196
0.0
76326
0.2
120
87856
16.8
1020
109231
45.3
Table 6:
Sixty-nine point initialization unsmoothed:
excess con centrat ion in the forward positive area
The excess was in about
initializations.
the
same
proportion
as
the
other
PAGE 109
v. Velocity of forward positive area
The
location
closely,
the
of
the
desired
forward
location
timestep 0010 the locatio n of the
area
(76+. 03147*T,26).
of
centroi d
area was at gridpoint (76.307,26).
of
predicted
value
of
the
positive
(76.3147,26).
very
close
At timestep 01 20, the centroid
was at gridpoint
(79.307 ,26).
gridpoint
(79.776,26),
prediction
by
predicted
about
that
positiv e
percent.
By
the
point.
By
location
was
lagged
the
1020,
the
area
t i m estep
,2
predicted
the positive area
pred i ct ed
location was at gridpoint (110. 1
area was mostly behind this
clear
0f
the
so
12
The
to
Th e centroid of the
positive area was, thus, only 2 percent be hind. the
location.
In
The ce ntr oid of the entire
cloud, including the wake was at (76.313,2 6),
the
very
approached,
6).
The positive
inspection,
centroid of the cloud still
it
lagge d about
percent behind the predicted location of the actual
was
ten
cloud.
vii. Conservation of mass
The graphs of the time history of the
area-weighted
of the concentrations is shown below in figure 16.
sum
PAGE 110
Figure 16:
Sixty-nine point initialization unsmoothed:
normalized mass
PAGE 111
The graph shows similar results as
this
case
the
systematically.
mass
in -the
model
previous
seemed
cases.
to
It would be hard to argue that
drop
the
In
more
drop
in
mass that was indicated is that first part of a random walk as
was possible with the nine point
was
initialization.
This
drop
the result of the accumulation of thousands of truncation
errors.
The
truncation
integer
rounding.
arithmetic
used
in
this
trial
used
The drop was very small, only amounting
to about a few percent of the total mass over the first day of
advection.
The drop appears to have been more smooth than the
variation in the other initializations.
This was probably due
to th' fact that the data was only sampled every ten timesteps
in this case and every timestep in the other,
With
rougher
cases.
this sampl ng frequency, random, short period variations
were, probably,
iltered
out
and
the
resulting
graph
was
dominated by the general downward trend.
viii. Conservation of squared mass
A graph showing the time history of the area weighted sum
of
17.
the squares of the concentrations is shown below in figure
PAGE 112
i18.00
Figure 17:
Sixty-nine point initialization unsmoothed:
normalized squared mass
PAGE 113
The growth of the sum of the squares of the mass in each
grid
box
was
initializations.
over
its
very
similar
to
that
in the
other
The quantity gre w slowly to about 60 percent
initial value.
This again represented a very small
average increase per timestep.
PAGE 114
2. Smoothed advection
a. Discussion
While the pure Arakawa scheme was
certain
properti es
of
a
well and it produiced quite
which
inevitabl y
designed
conserve
particle field, it did not do this
unrealis tic
dominated
the
re sults.
cloud
The
pattern
undesirable and spurious byproduct of this scheme.
wakes
were
In
an
order
to eliminate this problem, a method of smoo thing was developed
that prevented a wake from ever form ing at the lee side of the
cloud.
Very roug hily, the smoothing scheme prevented the first
wake from forming to the lee of the cloud.
It did this before
the negative wake grew large enough to form a positive wake to
its rear, and,
forming.
thus,
prevented
the
myri
of
wakes
It sub tracted the value o f the elim inated wake from
the concentration s in the rear of the cloud in order
mass conservation.
to
obey
The particular method of smoothing was not
standard and detaile d description of its mode
given in Appendix B.
b. One point initialization
i.
from
Description of initialization
of
action
is
PAGE 115
To compare the smoothed advection with the
advection,
the
were attempted.
10000
units
one, nine, and 69 po int
same
The one
of
point
concentration
trial
at
location was chosen to permit the
maximum
time
centered
because
wakes.
as
the
the gr id.
across
was
necessary
smoothing
was
gridpo
cloud
Arakawa
pure
nitial izations
in tialized
int
to
with
This
(10,26).
advect
for
the
The cloud did not have to be
wit h
the
unsmoothed advections
elim i nat ed the backwardly propagating
The smoothed trials, desc ribed here in sections c
and
d, were iterated for a much 1onge r time than was possible with
In the un smoothed
the unsmoothed trials.
would
reach
the
edge
of
the
timesteps.
The unsmoothed tr ials
timesteps,
stopping
only
boundary of the grid.
grid
trials,
in
lasted
the
wakes
approximately 1600
approximately
4200
when the cloud reached the eastern
Only the f irst day (1440
timesteps) of
advection has been considered in this paper.
ii.
Diagrams of advection
The results of the advection are shown below
18a through 18h.
30
0
0
0
0
0
0
9843
0
0
0
26
105
26
0
0
0
0
0
0
18a. On e point initial ization smoothed:
advection pattern at t=0001
in
figures
PAGE 116
0
0
0
0
23-~
0
0
0
9385
0
0
0
52
208
52
0
18b. One point initialization smoothed:
advection pattern at t=0002
0
0
0
2
-
0
0
0
0
9529
0
0
0
77
310
77
0
18c. One point initialization smoothed:
advection pattern at t=0 003
0
0
0
0
2 g-
0
0
0
0
0
8436
0
0
. 0
0
240
973
241
0
0
0~
0
18d. One point initialization smoothed:
advection pattern at t=0010
Zr-
0
0
0
0
0
0
0
0
0
0
405
0
0
0
0
0
180
1698
180
0
0
0
16
641
2039
643
16
0
0
102
588
1137
589
102
0
0
82
282
442
282
83
0
0
29
92
132
92
30
0
18e. One point initialization smoothed:
advection pattern at t=0100
PAGE 117
0
0
0
42
0
0
0
0
0
0
0
a- 0
0
0
0
0
0
281
0
0
0
0
8
264
776
261
8
0
0
130
51
955
583
128
0
0
203
0
187
425
557
427
188
0
54
524
598
207
0
0
122
250
315
252
124
0
0
57
121
149
123
60
0
2
1
41
5
441
2C
18f. One point initialization smoothed:
advection pattern at t=0300
0
0
25
24
25
0
0
0
0
0
0
if-
0
0
0
0
27
205
367
201
24
0
0
226
498
627
499
222
0
0
136
424
570
422
130
0
0
242
460
530
465
244
0
0
194
359
401
364
198
0
0
127
221
266
227
131
0
0
60
117
144
122
64
0
0
26
45
60
- 48
27
0
18g. One point initialization smoothed:
advection pattern at t=0600
ro
0
0
0
0
iS'-
0
1
0
00
6
0
0
0
12
0
0
93
145
144
146
93
0
178
291
311
292
180
0
0
0
228
377
414
380
233
0
0
242
400
445
405
250
0
0
222
376
419
381
231
0
0
188
312
344
317
195
0
120
213
243
218
128
0
0
62
117
142
117
62
0
18h. One point initialization smoothed:
advection pattern at t=1440
PAGE 118
iii. Description of advection
The results of the advection using the
of
the
Arakawa
scheme
unsmoothed trials.
were
much
more
smoothed
pleasing
After the early timesteps,
version
than the
when
most
of
the concentration was in the initialized grid box, the pattern
rapidly developed into an elliptical cloud.
The
model
cloud
exhibited features expected of a real cloud of particles.
The
concentration was highest in
and
dropped
monotonically
the
towards
center
of
the edges.
the
cloud
The cloud rapidly
grew to five gridpoints in length in the latitudinal direction
The
clouds
latitudinal
extent
remained
throughout the rest of the trial.
five
gridpoints
at five gridpoints
The longitudinal growth
was almost as rapid and the growth continued
slowly thereafter so that the cloud was 11 gridpoints long
the
end
of
the first day.
It
was
in
the
by
Apparently, the Arakawa Jacobian
strongly favors diffusion in the direction of
cloud.
to
direction
of
motion
motion
where
of
the
large
concentration gradients can coup le with large gradients in the
height field and produce a large Jacobian value at the edge of
the model cloud.
One unrealistic feature in the cloud
the
later timesteps.
trailers
noticeable
in
At the lee end of the cloud there were,
occasionally, trailing grid boxes with
If 'these
was
grew
too
small
concentrations.
long or separated from the the
PAGE 119
cloud, then the smoothing algorithm eliminated them.
some
trailers
timestep 1000.
they
were
not
were
still
However,
extant at the edge of cloud after
Since they contained only small concentrations
a
serious
problem
and
a
more
aggressive
smoothing algorithm could have eliminated them altogether.
iv. Cross-sections
Cross sections through the cloud
at
are shown below in figures 19a through 19d.
selected
timesteps
PAGE 120
LONGITUDE GRID#
19a. One point initialization smoothed:
cross-section at t=0100
PAGE 121
30.00
35.0
LONGITUDE GRIO#
19b. One point initialization smoothed:
cross-section at t=0300
PAGE 122
19c. One point initialization smoothed:
cross-section at t=0600
PAGE 123
19d. One point initialization smoothed:
cross-section at t=1440
PAGE 124
The cross-sections through the cloud
of
the
complicating structure was present without the smoothing.
The
cloud concentrations
distribution
quickly
formed
a
show
smooth
cloud.
bell
shaped
that advected downwind of its original location.
The peak concentration dropped as numerical
the
none
spread
diffusion
The peak values in the cloud were smaller than in
the unsmoothed cloud's forward positive area.
This
not
was
unexpected since the total concentration in the smoothed cloud
was also smaller
than
the
inflated
value
in
present
the
unsmoothed forward positive area.
v. Velocity of the.cloud
The cloud still moved more
first
this
slowly
than
the
wind.
difference was very large, but the cloud velocity
approached the wind velocity in the later timesteps.
showing
the
A
table
predicted longitudinal gridpoint location of the
centroid of the cloud and its actual location is
in table 7.
At
shown
below
PAGE 125
timestep
predicted location
of centroid
percentage lag
of centroid
actual location
of centroid
0001
10.03147
10.0157
50
0010
10.3147
10.1670
47
0100
13.147
12.34
26
1000
41.47
38.5
Table 7:
9
One point initialization smoothed:
lag in the centroid
While the centroid was not explicitly calculated for
unsmoothed
case, inspection will show that the smoothed cloud
was closer to the predi cted location than
was
in the unsmoothed trials.
positive
the
area,
alone,
properly interpreted as the "real
cloud".
that
unsmoothed
the
positi ve
trials
was
the
wake.
The centroid in the
The
figure 20.
The specific mechanism is
In this
in
lag
the
gridp oint
and
b
detailed
below
figure, the result of one time step of
advection on a single gridpoint is shown.
the
not have been
could
of the centro id of the smoothed cloud was due to the
smoothing process.
in
the
moved forward of the positive area by
the negati ve concentrat ions in
location
area
The lag in the centroid in the
forward po sitive area in the unsmoothed trials was due to
fact
the
a is the
value- at
is the value produced at a neighbor ing
gridpoint by the Jacobian.
PAGE 126
-
location
concentration
+1
t=0000
0
0
centroid
unsmoothed
advection
smoothed
advection
location
-1
0
+1
-1
concentration
-b
a
b
0
b
concentration
that
box
an
cloud
adjacen t
but the distan ce of this negative value (or reduced
reduced from one to zero.
figure
in
presente d
example
was
In
reduced.
20,
the
the
dist ance was
Thus the distance weighting used in
the centroid weighed this negative value less and
calculating
centroid
lagged
have
be hind
occup ie d.
why the centroid in thi s
gridpoint,
negative
woul d have helped move the cloud centroid
positive value) to the cloud centroid
otherwise,
the
removed
smoothing
The negative value was added to
forward.
t=0001
+b/a
the-
Essentially,
the
a-b
Cause of the centroid lag in smoothed advection
Figure 20:
simple
+1
+2*b/a
centroid
grid
0
had
the
location
that
it
would,
The example shown above explains
trial,
initialized
with
o nly
one
moved' only one half of the predicted distance
PAGE 127
in the first timestep.
cloud
became
larger
The
in
relative
lag
decreased
as. the
longitudinal extent and most of the
diffusion acted on internal and forward portions of the cloud.
In
these forward portions, the addition of the negative value
to a rearward gridpoint
having
a
did
not
negative concentration.
result
Thus,
in that
gridpoint
the smoothing scheme
was not activated and the advection scheme was not affected by
the
lag
mechanism
introduced by the elimination of negative
concentrations.
vi. Conservation of mass
The model preserved the mass of the cloud quite
is
used
shown
in
below
the
calculation.
in figure 21.
integer
arithmetic
well
as
Nearest integer rounding was
that
was
used
in
this
PAGE 128
Figure 21:
One point initial ization smoothed:
normalized mass
PAGE 129
The fluctuations in the mass of the cloud were much
erratic than in this trial's unsmoothed analogue.
oscillations
with
a
very
short
period
were
less
The regular
due
to
the
smoothing scheme.
vii. Conservation of squared mass
The time history of the area weighted sum of the
of the concentrations is shown below in figure 22.
squares
PAGE 130
Figure 22:
One point initialization smoothed
normalized squared mass
PAGE 131
The sum of the
of
in
mass
the
during the advection.
sharply
decreased
squares
gridbox
each
The decrease slowed
masses
with time and eventually the sum of the squares of the
became
reduction is
represents
The
constant.
nearly
shown
symbolically
in
quantity
the
a
below
23b.
figure
in
this
caused
that
process
a
grid box and b indicates the
quantity that was added to a neighboring grid box at the edges
of
the
Mass was conserved in these diffusion
cloud.
model
processes.
rear edge
unsmoothed trials:
location
concentration
forward edge
-1
0
0
+1
0
a
a
0
2*a
total sum of squares
nI
f
non wake diffusion
wake diffusion
location
-1
0
0
+1
concentration
-b
a+b
a-b
b
2
sum of squares
2
2
b + a + 2*a*b + b
total sum of squares
Figure 23a:
2*a*b + b
2
2
2*a
2
2
a -
+ 4*b
Cause of squared mass variation:
unsmoothed advection
+ b
PAGE 132
forward edge
rear edge
smoothed trials:
location
0
+1
concentration
a
0
2*a
total sum of squares
wake diffusion
non wake diffusion
0
location
a-b
concentration
sum of squares
of
2
2*a - 2*a*b + 2*b
diffusion.
the unsmoothed advection caused two
The
diffusion
wake
caused
concentration, a , to be spread between two grid boxes.
one
of the grid boxes was negative,
value of the sum of the squares of
type
of
+ b
Cause of squared mass variation:
smoothed advection
In the example above,
types
b
a - 2*a*b + b
total sum of squares
Figure 23b:
+1
diffusion
was
entirely
the
Since
this action increased the
the
masses.
The
second
positive and resulted in a
PAGE 133
the
decrease in the sum of the squares of
canceled and the resulting increase in the
almost
processes
sum of the squares of the masses was of order
was
the
value
two
The
masses.
b
squared.
b
that was diffused each timestep due to errors
was
and
approximations
difference
introduced by the finite
much smaller than a. Thus, the sum of the squared masses grew
slowly.
the
be
In the unsmoothed advection the wakes grew to
the main cloud, but only after there
as
magnitude
same
alternating
of
wakes
alternating
were several
of
which
sign
probably aided in canceling the growth of this.quantity.
When
the
smoothing
diffusion
was
eliminated
was
scheme
thus
and
unsmooth ed
more rapid than the growth in the
smoothed
were not adjacent to
Thus
absolutely.
in
grew
cloud
the
This
regi on
gr ew
decrease
of
rate
squares of the masses was
smaller
in
the
As
case.
of
thus,
the
relatively
and
in the sum of the
later
timesteps.
reduction in the decay rate of the su m of the squares is
shown in the lat er timesteps of figure 22.
very
squares
size, the number of grid boxes that
wake
the
the
was of order a*b and was,
decrease
This
masses.
wake
the unbalanced non-wake
diffusion acted alone and reduced the sum of
the
the
introduced,
high
in
thereafter.
timestep 1000.
the
The
first
quantity
100
was
The decay rate was
timesteps but dropped rapidly
virtually
constant
after
PAGE 134
viii. Discussion
The
improved
smoothed
results
version
of
the
advection
scheme
over the unsmoothed scheme.
The cloud that
was produced conserved mass and moved at an acceptable
The
main
flaw
in
gave.
speed.
the smoothed scheme was that the positive
numerical diffusion was not balanced by the wake particles and
thus
the
area
weighted
sum
of
the
concentrations decreased significantly during
This
decrease
squares
the
of
the
advection.
was an indicator of the scale of the numerical
diffusion introduced by finite differencing errors.
c. Nine point initialization
i.
Description of initialization
In an attempt to
simul ate
more
realistic
clouds,
the
multi-point initializations used in the unsmoothed trials were
repeated.
The
nine
point
initialization
was
centered
on
gridpoint (10,26).
ii. Diagrams of advection
The results of the advection are shown below
in
figures
PAGE 135
21a through 2t h.
0
0
1061
1079
zs7-
0
0
1109
1111
1108
0
0
1093
0
0
0
3
1126
1128
1126
3
0
24a. Nine point initial ization smoothed:
advection pattern at t=0001
10
2S-
0
0
1029
1045
0
0
1104
1110
1061
1105
0
0
0
1140
1145
1140
- 6
24b. Nine point initialization smoothed:
advection pattern at t=0002
00
0
0
1012
1014
-
1013
0
0
0
1100
1108
1101
0
0
9
1154
1162
1164
9
0
9
45
53
45
9
0
0
0
1
0
0
24c. Nine point initialization smoothed:
advection pattern at t=0003
0
0
804
799
r-. 8190
0
0
0
1053
1077
1055
0
0
0
25
1239
1271
1239
26
0
0
30
155
184
155
30
0
24d. Nine point initialization smoothed:
advection pattern at t=0010
PAGE 136
0
0
374
330
2Y- 380
0
0
0
125
879
1236
880
122
0
0
203
795
1033
798
210
0
0
155
413
530
414
158
0
0
68
155
195
156
69
0
0
18
43
54
43
19
0
0
0
4
8
4
0
0
24e. Nine point initialization smoothed:
advection pattern at t=0100
0
0
188
218
185
0
0
ZS-
0
121
523
747
518
117
0
0
228
613
791
614
226
0
0
253
490
613
493
254
0
0
173
321
384
322
177
0
0
87
168
199
169
88
0
0
33
70
83
70
34
0
0
1
18
23
18
2
0
24f. Nine point initialization smoothed:
advection pattern at t=0300
0
9
110
122
27- 107
4
0
0
106
345
461
343
98
0
0
201
473
573
474
193
0
0
248
474
547
477
249
0
0
221
392
453
397
225
0
0
153
266
317
279
157
0
0
85
157
184
159
90
0
0
39
72
87
73
40
0
0
6
22
26
22
7
0
24g. Nine point initialization smoothed:
advection pattern at t=0600
2S-
59
60
60
60
62
0
0
0
145
225
240
225
146
0
0
207
338
369
341
210
0
0
235
388
429
391
241
0
0
231
383
426
387
238
0
0
198
334
374
339
204
0
0
146
251
281
255
153
0
0
79
151
181
155
85
0
0
41
75
87
75
42
0
0
7
23
26
23
8
0
24h. Nine point initialization smoothed:
advection pattern at t=1440
PAGE 137
iii. Description of advection
T
concentrations in this trial formed
was ve
a
similar to the one formed in the previous trial.
major differ ence was during the early timesteps.
immediately
0f
in the rear.
also
expanded
distribution
and
more
become
elli ptical.
remained,
as
before
did
not
The expansion in the rear would have been
algorithm.
The
to the front and formed an arrow shaped
pointin g into the wind.
indistinct
The
pattern
the cloud, but this expansion
negative and was eliminated by the smoothing
cloud
The
that
to cover five latitudinal grid boxes in
expande d
the frontal portion
occur
pattern
and,
The
at
The arrow
became
more
by timestep 0100, the pattern had
latitudinal extent
a
constant
of
the
cloud
five grid boxes.
The
longitudinal extent also expanded at very nearly the same rate
as
in the previous trial.
The cloud was ten grid boxes long
in this direction by the end of one
day
of
advection.
The
noticeable impr ovement over the one point initialization
most
was the absence of the trailing particles at the edge
of
the
cloud.
iv. Cross sections
Cross sections through the cloud
at
are shown below in figures 25a through 25d.
selected
timesteps
PAGE 138
LONGITUOE GRIO#
25a. Nine point initialization smoothed:
cross-section at t=0100
PAGE 139
25b. Nine point initialization smoothed:
cross-section at t=0300
PAGE 140
25c. Nine point initialization smoothed:
cross-section at t=0600
PAGE 141
25d. Nine point initialization smoothed:
cross-section at t=1440
PAGE 142
After the early timesteps, the cloud appeared to be
similar
to the cloud produced in the previous initialization.
The peak values and shape were both almost exactly
The peak
value
was
slightly
smaller
than
positive area of the nine point unsmoothed
not
very
unexpected
since
the
peak
in
trial.
the
same.
the forward
This
was
value at initialization was
considerably smaller than in the previous trial.
v. Velocity of the cloud
The location of the centroid of
lagged
the
of
model
cloud
still
location that would have been predicted if it had
moved with the local wind velocity.
location
the
the
centroid,
is
The of
the
longitudinal
shown in figure table 8, for
selected timesteps.
timestep
predicted location
of centroid
actual -location
of centroid
percentage
lag
0001
10.03147
10.0206
35
0010
10.3147
10.259
18
0100
13. 147
12.80
11
1000
41.47
39.0
Table 8:
Nine point initialization smoothed:
lag in the centroid
8
PAGE 143
The lag was smaller than in the one point initialization,
especially
during
the
early
timesteps.
This was because a
relatively smaller portion of the cloud was bordering the wake
formation
region.
As with the previous trial, the percentage
lag decreased to a small value after 1000 timesteps.
vi. Conservation of mass
A graph of the area weighted sum of the concentrations is
shown below in figure 26.
PAGE 144
180.0s
Figure 26:
Nine point initial ization smoothed:
normalized mass
PAGE 145
The mass of the cloud was again well preserved.
integer
rounding
was
used
in this trial.
Nearest
The short period
artifact of the smoothing was also present.
vii. Conservation of squared mass
A graph of the area weighted sum of the
concentrations is shown below in figure 27.
squares
of
the
PAGE 146
TIME (MINUTES)
Figure 27:
(X10
Nine point initialization smoothed:
normalized squared mass
PAGE 147
in the one gridpoint initialization.
than
trial
The squared mass dropped more slowly in this
This was because some of
the gridpoints were removed from the high gradient regions
lee
the
of the cloud where the action of the smoothing
side
timestep
and
0400
began
then
scheme
off to an,
level
The oscillation caused by the
apparently, steady state value.
smoothing
to
until
sharply
dropped
The squared mass
scheme was evident.
about
at
had. become a noticeable factor in the drop
rate of this quantity.
however,
small,
This oscillation was
compared to the systematic drop in the squared mass.
d. Sixty-nine point initialization
i. Description of initialization
with
The 69 point initialization was also attempted
scheme.
smoothed
distribution was at
from
shift
the
The
gridpoint
central
location
the
edge of the grid.
of
of
the
zero
in
this
region.
gradient of height val'ues were
other
avoiding
the
smoothed
region
The height field produced by the
interpolation scheme described in Appendix A
to
gridpoint
one
The
(11,26).
initializations was for the purpose
near
the initial concentration
of
center
the
The
winds
dropped
rapidly
implied by this high
unrealistic
and
were
to
be
PAGE 148
avoided.
ii.
Diagrams of advection
The results of the advection are shown below
in figures
26a through 28h.
2.V-
0
0
0
1
1
2
1I
1
0
0
0
0
0
0
13
66
111
66
13
0
0
0
0
15
197
916
1523
915
197
15
1
0
0'
0
2
4
76! 133
951 1620
4307' 7259
7118 11969
4306 7259
950, 1620
76; 133
4
2
0
0
0
2
86
1013
4499
7400
4500
1014
86
2
0
0
1
21
241
1048
1717
1049
241
21
1
0
0
0
2
23
96
155
96
23
2
0
28a. Sixty-nine point initialization smoothed:
advection pattern at t=0001
0
0
1
0
0
z%'-
0
1
0
0
1
12
175
851
1427
850
175
12
1
4
2
133
71
919 1619
4209 7256
6975 11964
4208 7256
918, 1619
711 133
4
21
2
91
1044
4594
7540
4596
1045
91
2
1
24
263
1116
1816
1117
263
25
1
-7?
0
3
28
112
178
112
28
3
0
28b. Sixty-nine point initialization smoothed:
advection pattern at t=0002
0
0
1
0
0
.r-
0
1
0
0
1
9
153
787
1333
786
153
9
1
4
2
132
66
8871 1617
4111! 7250
683111956
4109 7250
886 1617
132
66
4
2
2
96
1075
4688
7679
4691
1076
96
2
1
28
286
1185
1917
1186
286
29
1
,f
0
4
34
129
203
129
34
4
0
28c. Sixty-nine point initialization smoothed:
advection pattern at t=0003
PAGE 149
7$-
O
0
1
0
0
0
1
0
0
zS-
0
0
0
0
0
0
0
0
0
2
4
121
31
644
1567
3400 7128
5793 11783
3396 7126
642 1566
31,
121
2'
4
1
0
25
363
642
360
25
0
1
2
125
1274
5314
8601
5318
1276
125
2
1
56
457
1696
2664
1702
458
57
1
0
13
86
276
416
277
86
13
0
0
0
9
26
38
27
10
0
0
28d. Sixty-nine point initialization smoothed:
advection pattern at t=0010
2
0
0
1
0
0
0
1
0
0
2-
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
21
0
0:
0
0
0
0
0
2
4
0
0
0
0
0
0
0
4
0
0
0
2408
2466
2338
0
0
0
0
7
1136
5805
9269
5791
1120
2
0
0
214
1782
5388
7736
5405
1789
213
0
y
0
279
1281
3094
4159
3111
1297
281
0
13
180
621
1305
1675
1317
632
182
14
2
75
228
436
540
439
232
76
3
0
20
64
118
142
118
65
21
0
0
0
12
23
29
23
12
0
0
28e. Sixty-nine point initialization smoothed:
0
0
0
1012
1005
2C -1011
0
0
0
0
0
715
3344
S021
3324
699
0
0
0
179
1641
4318
5679
4329
1626
166
0
51
461
1826
3630
4513
3649
1842
463
48
103
507
1374
2404
2891
2428
1392
523
103
88
362
830
1342
1578
1360
850
377
89
61
201
425
650
755
660
436
210
62
.
?
advection pattern at t=0100
33
93
187
279
320
282
192
96
34
4
28
66
103
117
104
69
30
6
0
0
16
26
33
27
17
0
0
28f. Sixty-nine point initialization smoothed:
advection pattern at t=0300
2-
0
0
0
0
334
0
206 1655
189 2520
214 1621
308
0
0
0
0
0
0
126
1163
2925
3732
2920
1122
101
0
65
503
1740
3146
3739
3157
1739
490
57
173
728
1687
2699
3147
2723
1714
732
167
231
674
1338
1985
2272
2007
1366
691
235
195
505
914
1296
1461
1317
940
523
203
135
321
555
765
854
780
576
335
143
74
190
305
412
456
420
318
190
82
36
83
146
200
219
205
154
90
39
5
27
57
81
90
84
61
30
9
0
0
12
22
24
23
14
0
0
28g. Sixty-nine point initialization smoothed:
advection pattern at t=0600
I
0
0
0
105
366
374
379
69
0
0
102
616
1279
1513
1248
539
57
0
23
415
1189
1843
2093
1326
1135
353
0
147
733
1437
2030
2275
2036
1447
696
112
267
834
1430
1928
2127
1953
1463
842
244
335
794
1267
1654
1808
1684
1305
821
328
302
673
1024
1304
1419
1335
1065
707
318
237
511
761
955
1036
982
803
546
254
173
347
523
654
710
674
560
305
190
108
219
334
416
452
431
359
249
126
57
120
190
240
265
252
210
141
64
25
54
90
123
135
129
102
66
32
28h. Sixty-nine point initialization smoothed:
advection pattern at t=1440
0
15
29
46
53
50
34
21
0
0
0
0
6
9
8
1
0
0
PAGE 150
iii. Description of advection
to the other initializations.
The latitudinal extent of the cloud remained constant
trials.
extent
longitudinal
They arose at the lee edge
present.
edges
regions
from
slowly
of one day.
end
the
by
gridpoints
grew
of
of
low
very
initialization.
cloud.
the
The
initialization.
the
of
gridpoints
nine
the
south
The pattern did not exhibit any
spreading that was evident in the previous
immediate
the
at
a
of considerable extent, was qualitatively quite similar
cloud
of
as
initialized
The advection pattern, in this trial,
to
nine
fourteen
The trailers were again
and
the
at
north
and
They formed because there were
gradients
of
in
concentration
the
The concentrations in these regions were not
Any
advected and were left behind as trailers.
advection
in
this region was lost to the truncation error introduced by the
integer arithmetic.
primitive
version
This trial was
have
a
with
more
of the smoothing algorithm which could not
eliminate all of the trailers.
they
calculated
In order
to
conserve
space,
been left off of the above diagrams when the cloud
has 1eft them far behind.
iv. Cross-sections
Cross sections th'rough the cloud
at
selected
timesteps
PAGE 151
are shown below in figures 29a through 29d.
76.00
82.00
88.00
94.00
100.00
106.00
112.00
118.00
124.00
LONGITUDE GRID#
29a. Sixty-nine point initialization
cross-section at t=0100
smoothed:
PAGE 152
94.00
100.00
LONGITUDE GRID#
29b. Sixty-nine point initialization smoothed:
cross-section at t=0300
PAGE 153
29c. Sixty-nine point initialization smoothed:
cross-section at t=0600
PAGE 154
94.00
100.00
106.00
112.00
LONGI TUDE GRID#
29d. Sixty-nine point initialization smoothed:
cross-section at t=1440
PAGE 155
v. Velocity of the cloud
The lag in the
elimination
of
location
the
of
the
cloud
caused
by
negative wake was sharply reduced.
the 69 point initialization, the values in the
the
With
negative
wake
that were eliminated were very small when compared to the main
cloud values.
A table showing the lag at
selected
timesteps
is shown below in table 9.
predicted location
of centroid
timestep
actual location
of centroid
percentage
lag
0001
76.03147
76.03125
0.7
0010
76.3147
76.311
1
0100
79.147
78.953
6
106.14
107.47
1000
Table 9:
Six ty-nine point initialization smoothed:
lag in the centroid
The lag in the early timesteps
value
to
4
a
few percent.
grew
from
a
negligible
The reason for this growth was the
elimination of the buffer of small concentrations at
end
of
end
lee
the cloud that suppressed wake formation and thus the
lags that were due to wake elimination.
lee
the
By timestep 0100, the
of the cloud contained large concentrations and wake
elimination had noticeable effects on the speed of the
cloud.
PAGE 156
As
before,
as the cloud spread in the 1ongitudinal direction,
the fraction of grid boxes exposed
shrank
and
the
to
the
wake
elimination
cloud began to attain the predicted velocity
and close the percentage gap between its actual and
predicted
locations.
vi. Conservation of mass
A graph of the area weighted sum of the concentrations is
shown below in figure 30.
PAGE 157
180. 00
Figure 30:
Sixty-nine point initial ization smoothed:
normalized mass
PAGE 158
The mass in thi s trial was lost slowly through truncation
error .
The
model
in this trial used truncation rounding in
some of its integer arithmetic.
However, the
mass
is
still
fairl y well conserve d.
vii. Conservation of squared mass
A graph detailing the time history of the
sum
of
the
figure 31.
squares
area
weighted
of the concentrations is shown below in
PAGE 159
TIME (MINUTES)
Figure 31:
(X10
1
)
Sixty-nine point initialization smoothed:
normalized squared mass
PAGE 160
The squared mass decreased,
trials, rapidly at first.
state
of
initialization.
trailers
trial.
by
the
in
the
other
smoothed
The rate of decrease became smaller
in later timesteps and the value
steady
as
about
one
approached
fifth
an,
of
apparently,
the
value
at
The oscillations caused by the elimination of
smoothing
scheme were very evident in this
They were still small compared to the general downward
trend in the squared mass.
PAGE 161
3. Trial with deposition
a. Discussion
The eventual goal of this advection model is
of
amount
the
fallout
such
deposition,
predict
would reach the ground after a
In order to test the
nuclear explosion.
for
that
to
model's
predictions
a trial that incorporated a scheme for
this deposition was attempted.
b. Description of deposition scheme
in- this
The deposition scheme that was used
was
trial
very crude.
in a grid box
were
diagnostic
The scheme assumed that the particles
uniformly
distributed
with
respect
up to the level at which the advection was calculated.
height
In this particular case, the concentration
assumed
falling
of
particles
was
to have been constant from the ground up to the 30000
The scheme then assumed that the particles were all
Pa level.
at a single mean terminal velocity, v .
In a certain
amount of time, all particles in the lowest v,* 4t
the
to
cloud
fell and were deposited on the ground.
meters
of
The scheme
then assumed that the concentration field instantly readjusted
so
This
that
the
concentration
assumption
essentially
was again independent of height.
def ined
an
infinite
vertical
PAGE 162
diffusion constant.
The process just described is detailed
the diagram below in figure 32.
Vf at
____
4'IIt
Vep A+/I
CROVND
Figure 32:
Process of deposition
While this deposition scheme made some rather unrealistic
assumptions,
it
lead immediately to a simple modification of
the advection equations that permitted direct
the
deposition on the ground.
for the drpIn
If (16)
concen'tration after
the
computation
of
is modified to account
readustment
of
the
PAGE 163
concentration field, the result is:
dC
DC
g P
-- - ---------------------------J(Cz)
dt
2 om sin(th) Re p cos(th) r T
Dt
where H is the vertical depth of the cloud.
v? C
(36)
H
Computationally,
this modification just indicates that a certain fixed fraction
of the concentration in a grid box is deposited on the
each
timestep.
Thus
the
ground
concentrations in the model cloud
should tend to decay in an exponential manner..
c. Description of initialization
The cloud was initialized using the 69 point
concentrations.
The
center
of
the
pattern
of
cloud was at gridpoint
(11,26)
d. Deposition parameters
The model removed a fraction of the concentration in each
grid
box
each timestep and placed it in a corresponding grid
box representing the ground.
the
last
term
in
the
The fraction was
equation
above.
determined
by
The values of the
cons tants were set at H=9300 meters which was a typical he ight
PAGE 164
value
in
the
height field, and v =.48 in s-1.
the velocity was taken
from
Glasstone
The value for
(1977).
It
is
the
approximate value for the particles that carry the bulk of the
radioactivity from an explosion.
Thus,
in
(C)*(.48m s-1)*(60s timestep-1)/(9300
in)
on the ground and .997*C remained
the
advected.
in
=
every
.003*C
timestep,
was placed
atmosphere
to
be
The model used integer arithmetic to calculate the
remaining concentration in the cloud and used
floating
point
The
floating
point
because
many
arithmetic
to
calculate
deposition.
arithmetic was used for the deposition
deposited
amounts
were
small
compared
to
the
time
the
and the computational cost in
time caused by the use of the floating
small
of
required
point
for
arithmetic
was
the advectional
calculations.
e. Diagrams of advection
The results of the advection are shown below in figures
33a through 33h.
10
I0
25 -
0
0
1
1
2
1
1
0
0
0
0
13
66
111
66
13
0
0
1
15
196
913
1518
912
196
15
1
2
4
76
133
949 1615
4294 7237
7097 11932
4293 7237
947 1615
76
133
2
4
2
86
1010
4486
7378
4487
1011
86
2
1
21
240
1045
1712
1046
240
21
1
0
2
23
96
155
96
23
2
0
33a. Trial with deposition:
advection pattern at t=0001
0
0
1
3
6
3
1
0
0
PAGE 165
1o
2 S-
0
0
1
0
0
0
1
0
0
0
0
9
52
90
52
9
0
0
1
12
173
845
1419
844
173
12
1
2
4
71
133
913 1609
4184 7212
6934 11992
4182 7212
912 1609
71
133
2
4
2
91
1033
4567
7495
4568
1039
91
2
1
24
261
1109
1806
1111
261
24
1
0
3
28
112
177
112
28
3
0
0
0
1
4
9
4
1
0
0
33b. Trial with deposition:
advection pattern at t=0002
2s-
0
0
1
0
0
0
1
0
0
0
0
5
39
68
39
5
0
0
1
9
152
779
1322
778
151
9
1
2
4
66
132
878 1602
4074 7184
6771 11848
4072 7184
877 1602
66
132
2
4
2
96
1065
4647
7610
4648
1067
96
2
1
28
282
1173
1900
1176
283
28
1
0
4
34
129
201
129
34
4
0
0
0
2
6
10
6
2
0
0
33c. Trial with deposition:
advection pattern at t=0003
2i -
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
26
352
625
351
25
0
1
4
2
31
121
625 1518
3299 6916
5623 11439
3294 6919
623 1517
31
121
2
4
2
125
1236
9157
8347
5161
1238
125
2
1
56
443
1645
2586
1651
444
56
1
0
13
89
271
404
271
86
13
0
0
0
9
26
38
26
9
0
0
33d. Trial with deposition:
advection pattern-at t=0010
1
0
0
0
0
2'- 0
0
0
1
2
0
0
0
0
0
0
0
2
4
0
0
0
0
0
0
0
4
2
0
0
1781
1831
1769
0
0
2
0
20
854
4294
6866
4278
844
18
0
169
1319
3992
5733
4005
1332
166
216
950
2291
3087
2306
960
218
145
464
967
1244
974
472
150
61
179
327
406
328
182
62
33e. Trial with deposition:
advection pattern at t=0100
17
53
9S
116
97
54
17
0
9
18
23
18
9
0
PAGE 166
ar-
0
0
402
402
0
293
1369
2044
103
724
1774
2308
195
749
1479
1839
195
556
973
1173
161
339
542
636
95
192
271
311
42
93
138
155
408
1363
1780
1496
981
548
274
139
0
0
279
0
715
98
755
191
560
195
343
164
195
98
95
44
33f. Trial with deposition:
advection pattern at t=0300
30
0
2-
0
35
32
36
0
0
32
83
269
421
261
71
27
114
264
488
629
489
258
109
165
300
538
646
540
302
163
166
261
440
530
446
265
166
150
192
288
334
293
197
152
117
166
180
196
181
166
119
84
141
166
166
166
145
86
51
96
126
135
127
100
53
33g. Trial with deposition:
advection pattern at t=0600
31
0
0
0
Ar-
0
0
29
77
82
82
67
91
83
90
87
115
139
134
139
116
91
84
122
166
166
166
124
89
92
121
166
166
166
122
89
90
120
166
166
166
122
86
79
119
166
166
166
121
85
77
118
165
166
166
120
83
33h. Trial with deposition:
advection pattern at t=1440
76
105
157
166
161
108
82
45
72
123
144
127
75
50
PAGE 167
f. Description of advection.
The advection pattern was very similar
the
of
advection
69
point
latitudinal
trailers
when
concentration
not fill these
later
During
the
left
were
failure
behind.
(see
section
of
the
cloud
to
The
total
smaller
in this cloud was the reason why this trial did
layers
where
the
nondepositing
did.
trial
timesteps, the cloud began to develop a line of
to
enough
not
The smoothing scheme was
trailers that were left behind.
aggressive
ground.
northern and southern layers that were left empty
the
refill
major
extent of the cloud decreased to seven
This was due to
gridpoints.
The
the
mass
timesteps, the cloud had lost most of its
The
to
particles
smoothed.
decreased significantly, and by the later
concentrations
h).
the
initialization.
difference was due to the loss of
The
to
eliminate
these lines and, thus, they
persisted and formed a long trail behind the main cloud.
The most notable problem in the advection pattern was the
end of particle loss due to deposition in the later timesteps.
The concentrations in the center of the
below
166
attained
units
this
cloud
did
fall
not
and eventually a large number of grid boxes
same
minimum
value.
This
problem
can
be
attributed directly to the integer arithmetic that was used to
calculate the concentrations that remained in the cloud
after
PAGE 168
the
deposition.
When
a
grid
box
166 units of
contained
units.
concentration, the resulting deposition was .498
remaining
cloud
concentration
was corrected to 166--
165.502 units, which was rounded up to 166 units.
concentration
never
could
The
.498 =
Thus,
the
never drop below 166 units except
through numerical diffusion in
the
advection
scheme.
This
diffusion was a very slow process in the later timesteps.
g. Cross-sections
Cross sections through the cloud are shown
timesteps in figures 34a through 34d.
for
selected
PAGE 169
30.00
35.00
LONGITUDE GRID#
34a. Trial with deposition:
cross-section at t=0500
PAGE 170
.00
25.00
30.00
35.00
40.00
45,
LONGITUDE GRID#
34b. Trial with deposition:
cross-section at t=0600
PAGE 171
34c. Trial with deposition:
cross-section at t=1000
PAGE 172
60.00
34d. Trial with deposition:
cross-section at t=1440
PAGE 173
These cross sections show a cloud that was
in
shape
to
units.
similar
The exception was
the case with no deposition.
the development of the cutoff that was
very
at
concentration
166
This flat region was kept high by the rounding errors
just described in section f.
h. Conservation of mass
The deposition scheme removed a
fraction
fixed
of
As was
concentration from each grid box during each timestep.
expected,
this
concentrations
caused
to
sum
the
drop
of
exponentially
the
area
un til
the
weighted
integer
truncation error halted the loss of particles from the
From that point on, mass was almost perfectly conserved.
pattern is shown in figure 35.
the
cloud.
This
PAGE 174
Figure 35:
Trial with deposition:
normalized mass
PAGE 175
i.
Conservation of squared mass.
The area
decreased
weighted
rapidly.
This
loss due to deposition.
was
approached
state
of
was
the
squared
concentrations
a result of diffusion and the
The almost steady
state
value
that
in the later timesteps was about three orders
of magnitude lower than
steady
sum
the
value
at
initialization.
The
was reached as diffusion and deposition stopped
in the later timesteps.
This pattern is shown in figure 36.
PAGE 176
Figure 36:
Trial with deposition:
normalized squared mass
PAGE 177
j. Diagrams of deposition
The pattern of deposition
on
the
ground
at selected
timesteps is shown below in figures 37a through 37 i.
0.0
0.0
0.2
0.3
- 0.2
0.0
0.0
2
0.0
0.6
2.7
4.6
2.7
0.6
0.0
0.2
2.9
12.9
21.4
12.9
2.0
0.2
0.4
4.9
21.8
35.9
21.8
4.9
0.4
0.3
3.0
13.5
22.2
13.5
3.0
0.3
0.1
0.7
3.1
5.2
3.1
0.7
0.1
0.0
0.1
0.3
0.5
0.3
0.1
0.0
37a. Trial with deposition:
deposition pattern at t=0001
I#
0.0
0.1
0.4
0.6
0.4
0.1
0.0
25-
0.1
1.1
5.3
8.8
5.3
1.1
0.1
0.4
5.6
25.5
42.2
25.5
5.6
0.4
0.8
9.7
43.5
71.7
43.5
9.7
0.8
0.5
6.2
27.2
44.8
27.2
6.2
0.5
0.1
1.5
6.5
10.6
6.5
1.5
0.1
0.0
0.2
0.6
1.0
0.6
0.2
0.0
37b. Trial with deposition:
deposition pattern at t=0002
0.0
0.1
0.5
0.8
jy
-
0.5
0.1
0.0
0.1
1.6
7.6
12.8
7.6
1.6
0.1
1.2
0.6
8.2 14.5
65.1
37.8
62.6 107.3
65.1
37.8
14.5
8.2
1.2
0.6
0.8
9.4
41.2
67.7
41.2
9.4
0.8
0.2
2.4
10.0
16.3
10.0
2.4
0.2
0.0
0.3
1.0
1.6
1.0
0.3
0.0
0.0
0.0
0.0
0.1
0.0
0.0
0.0
37c. Trial with deposition:
deposition pattern at t=0003
I
0.1
0.1
3.9
3.2
0.1
1.6
3.2 23.7 47.3 33.9
18.8 114.3 213.4 145.4
32.5 191.4 352.3 237.0
18.7 114.2 213.4 145.5
47.3
34.0
3.2
23.7
1.6
3.9
3.2
0.1
0.1
0.1
0.1
0.1
25s-
0.1
0.6
1.1
0.6
0.1
1.1
10.2
40.3
64.5
40.5
10.2
1.1
0.2
1.5
5.4
8.2
5.4
1.6
0.2
37d. Trial with deposition:
deposition pattern at t=0010
0.1
0.4
0.6
0.4
0.1
0.0
0.0
0.3
0.0
0.0
z-
0.0
0.3
0.0
0.0
0.0
0.0
0.1
0.6
1.1
0.6
0.1
0.0
0.0
0.0
0.2
0.6
1.2
0.6
0.3
8.9 22.6 35.3 36.9
1.8
0.1
40.0 164.1 291.1 309.3 209.1
3.2
22.1 259.6 975.41449.61125.4'587.3
38.4 422.51576.92358.51722.2 833.0
22.0 259.1 974.61447.61128.2 589.6
3.2
39.9 163.4 289.2 309.9 210.4
8.9 22.? 35.1 36.9
1.8
0.1
0.0
0.2
0.6
1.2
0.3
0.6
0.0
24.8
99.9
234.2
314.6
235.8
100.8
25.3
0.0
0.0
11.0
36.9
76.3
97.9
76.6
37.5
11.5
0.0
0.0
0.5
2.3
4.6
5.8
4.7
2.3
0.5
0.0
0.0
3.4
11.0
20.8
26.0
20.9
11.1
3.4
0.0
0.0
0.0
0.1
0.5
0.7
0.5
0.1
0.0
0.0
37e. Trial with deposition:
deposition pattern at t=0100
0.0
0.0
0.3
0.0
0.0
0.0
0.3
0.0
0.0
2S-
0.0
0.0
0.1
0.6
1.1
0.6
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.2
1.8
3.6
1.8
0.9
65.7 70.6
0.1
8.9 22.6 35.4 48.2 57.5
1.8
40.0 164.1 291.1 352.8 371.7 378.0 373.4 360.7
3.2
22.1 259.6 975.41501.41501.11356.71206.31073.3 948.4
39.4 422.51576.92407.82320.81996.91715.2 1482.81278.9
22. 0 259.1 974.61501.31502.41357.51210.51077.5 950.1
39.9 163.4 289.2 352.2 374.1 377.7 372.9 361.9
3.2
8.9 22.7 35.2 47.6 57.7 66.6 70.5
1.8
0.1
0.0
0.0
0.0
0.0
0.2
1.8
3.6
1.8
0.9
0.0
76.6
329.5
756.5
982.8
762.0
330.5
76.3
0.0
0.0
75.0
261.3
513.5
638.5
517.6
264.0
75.3
0.0
0.0
0.0
61.8
38.5
169.0
95.2
299.7 155.4
363.5 183.9
303.7 156.8
171.2
96.2
62.5
39.3
0.0
0.0
0.0
21.2
47.4
72.7
84.2
73.5
48.0
21.9
0.0
37f. Trial with deposition:
deposition pattern at t=0300
z0
0.0
0.0
0.3
0.0
0.0
1-
0.0
0.3
0.0
0.0
0.0
0.0
0.1
0.6
1.1
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.2
3.6
7.2
3.6
1.8
8.9 22.6 35.4 48.2 57.5 65.7 70.6 76.6
1.8
0.1
3.2 40.0 164.1 291.1 352.8 371.7 378.0 373.4 360.7 340.2
22.1 259.6 975.41501.41501.11356.71206.31073.3 957.7 851.4
38.4 422.51576.92407.82320.81996.91715.21482.81287.31127.5
22.0 259. 1 974. 61501. 31502. 41357. 51210.51077.5 959. 6 859.8
39.9 163.4 289.2 352.2 374.1 377.7 372.9 361.9 340.1
3.2
8.9 22.7 35.2 47.6 57.7 66.6 70.5 76.3
1.8
0.1
0-0
0.0
0.0
0.0
0.0
0.2
3.6
7.2
3.6
1.8
0.0
83.2
266.5
547.7
682.0
zS - 550.4
266.7
83.4
0.0
0.0
84.8
242.5
489.2
601.4
492.0
242.7
85.0
0.0
0.0
85.4
231.9
434.3
537.5
436.3
231.7
86.1
0.0
0.0
87.4
216.9
393.7
472.3
394.8
216.2
87.4
0.0
0.0
0.0
0.0
50.1
66.5
79.9
85.3
164.1 118.9
263.2 196.0 115.5
309.9 214.8 133.7
265.6 187.5 117.0
87.0
165.4 120.5
51.1
67.5
80.5
0.0
0.0
0.0- _0.0
0.0
89.7
202.3
335.5
405.2
337.8
202.4
90.2
0.0
34.0
55.8
72.9
78.6
73.6
57.0
34.9
0.0
0.0
20.7
37.1
45.5
49.7
45.7
38.1
21.4
0.0
0.0
80.2
327.3
763.6
986.7
766.6
327.9
80.0
0.0
0.0
85.2
308.3
680.0
869.2
685.5
308.7
84.8
0.0
0.0
10.7
20.8
27.9
30.1
28.1
21.7
11.2
0.0
37g. Trial with deposition:
deposition pattern at t=0600
0.0
4.0
8.6
13.1
14.7
13.2
9.1
4.3
0.0
0.0
e2.4
287.9
607.7
765.3
610.9
28.6
82.1
0.0
0.0
0.5
2.2
3.8
4.5
3.8
2.3
0.6
0.0
0.0
0.0
0.0
0.4
0.5
0.4
0.1
0.0
0.0
0.0
8.9
21.2
31.6
36.0
31.8
21.8
9.2
0.0
0.0
2.7
7.3
12.0
13.5
12.1
7.6
3.0
0.0
0.0
0.2
1.5
3.0
3.6
3.1
1.6
0.3
0.0
0.0
0.0
0.0
0.2
0.3
0.2
0.0
0.0
0.0
I0
0.0
0.0
0.3
0.0
0.0
2s. -0.0
0.3
0.0
0.0
0.0
0.0
0.1
0.6
1.1
0.6
0.1
0.0
0.0
2S-
0.1
3.2
22.1
38.4
22.0
3.2
0.1
3.0
0.0
90.5
204.7
349.3
425.9
351.3
204.3
90.7
0.0
0.0
0.0
0.0
0.0
0.0
0.2
6.0
6.0 12.0
65.7 70.6 76.6
8.9 22.6 35.4 48.2 57.5
1.8
40.0 164.1 291.1 352.8 371.7 378.0 373.4 360.7 340.2
259.6 975.41501.41501.11356.71206.31073.3 957.7 854.4
422.51576.92A 07.82D20.81996.91715.21482.81287.31127.5
259.1 974.61501.31502.41367.51210.51077.5 959.6 859.8
39.9 163.4 289.2 352.2 374.1 377.7 372.9 361.9 340.1
1.8.9 22.7 35.2 47.6 57.7 66.6 70.5 76.3
0.0
0.0
0.0
0.0
0.0
0.2
6.0
6.0 12.0
0.0
89.7
186.0
314.0
379.5
315.7
185.8
89.4
0.0
jr-
0.0
93.6
166.5
293.5
336.4
283.9
166.4
93.6
0.0
0.0
96.3
154.0
255.7
294.8
257.0
154.5
96.2
0.0
0.0
55.0
70.5
92.9
97.1
0.0
42.1
56.1
75.9
80.7
76.7
57.3
44.1
0.0
93.6
72.4
57.0
0.0
0.0
99.8
141.4
235.8
263.9
236.5
140.8
98.8
0.0
0.0
30.0
41.6
59.1
63.9
60.3
42.5
31.7
0.0-
0.0
0.0
0.0
105.5 103.1 1 02.7
120.9 123.0 1 26.6
212.6 184.2 1 59.1
249.9 222.9 1 99.5
213.2 185.6 1 58.4
123.8 122.0 1 26.9
103.4 104.8 e 02.8
0.0
0.0
0.0
0.0
18.6
27.5
42.7
47.5
43.3
28.8
20.0
0.0
0.0
9.0
15.8
26.1
30.5
27.0
16.4
10.0
0-0
0.0
2.8
7.0
12.7
15.2
12.8
7.7
3.4
0.0
--
0.0
104.9
120.7
141.4
167.8
141.2
129.2
102.3
0.0
0.0
106.7
120.7
140.9
146.1
141.6
130.3
102.9
0.0
0.0
0.1
1.9
3.6
4.6
3.8
2.1
0.3
0.0
37h. Trial with deposition:
deposition pattern at t=1000
0.00.0
0.0
0.4
0.6
0.4
0.0
0.0
0.0
0.0
0.0
85.2
80.2
327.3 308.3
763.6 680.0
996.7 669.2
766.6 685.5
327.9 308.7
84.0
80.0
0.0- 0.0
0.0
104.5
119.8
140.3
144.7
141.8
121.4
104.0
0.0
0.0
96.1
114.7
136.2
10.2
137.2
116.6
96.7
_0-0
0.0
82.4
287.9
607.7
765.3
610.9
288.6
82.1
0.0
0.0
83.2
266.5
547.7
632.0
550.4
266.7
03.4
0.0
0.0
0.0
83.1
68.6
86.1
101.4
125.3 109.3
129.0 113.8
126.1 110.5
87.4
104.1
85.0 70.9
320
-- 2L.
0.0
8A.8
242.5
489.2
601.4
492.0
242.7
85.0
0.0
0.0
85.4
231.9
434.3
537.5
436.3
231.7
86.1
0.0
0.0
87.4
216.9
394.2
472.5
395.2
216.2
87.4
0.0
0 0
0.0
0.3
o 0
0.0
2-
0.0
0.3
0.0
0.0
0.0
0.1
0.6
1.1
0.6
0.1
0.0
0.0
0.0
0.0
0.2
6.0
6.0 12.0
3.0
8.9 22.6 35.4 48.2 57.5
1.8
0.1
3.2 40.0 164.1 291.1 352.8 371.7 378.0
22.1 259.6 975.41501.41501.11356.71206.3
38.4 422.51576.92407.82320.81996.91715.2
22.0 259.1 974.61501.31502.41357.51210.5
39.9 163.4 289.2 352.2 374.1 377.7
3.2
8.9 22.7 35.2 47.6 57.7
1.8
0.1
0.0
0.0
0.2
6.0
6.0 12.0
3.0
0.0
96.3
154.0
255.7
294.8
257.0
154.5
96.2
0.0
0.0
90.5
216.9 204.7
294.2 349.3
472.5 425.9
zr- 395.2 351.3
216.2 204.3
90.7
87.4
0.0
0.0
0.0
99.7
186.0
314.0
379.5
315.7
185.8
89.4
0.0
0.0
93.6
166.5
283.5
336.4
283.9
166.4
93.6
0.0
0.0
111.0
111.7
141.7
144.3
2S- 142.1
113.5
113.5
0.0
0.0
100.3
109.6
142.1
144.4
142.2
111.9
103.5
0.0
0.0
0.0
86.3
93.6
109.5 110.7
142.0 142.7
144.5 144.4
142.6 143.1
111.7 111.9
91.2
97.0
0.0- 0.0
0 .0
0.0
105.4
111.0
141.7
143.9
142.6
112.4
108.2
0.0
0.0
0.0
0.0
65.7 70.6 76.6
373.4 360.7 340.2
1073.3 957.7 854.4
1482.81287.31127.5
1077.5 959.6 859.8
372.9 361.9 340.1
66.6 70.5 76.3
0.0
0.0
0.0
0.0
80.2
327.3
763.6
986.7
766.6
327.9
80.0
0.0
0.0
85.2
308.3
680.0
869.2
685.5
308.7
84.8
0.0
0.0
82.4
287.9
607.7
765.3
610.9
288.6
82.1
0.0
0.0
83.2
266.5
547.7
682.0
550.4
266.7
83.4
0.0
0.0
84.8
242.5
489.2
601.4
492.0
242.7
85.0
0.0
0.0
85.4
231.9
434.3
537.5
436.3
231.7
86.1
0.0
0.0
99.8
141.4
235.8
263.9
236.5
140.8
98.8
0.0
0.0
105.5
120.9
212.6
249.9
213.2
123.8
103.4
0.0
0.0
103.1
123.0
184.2
222.9
185.6
122.0
104.8
0.0
0.0
102.7
126.6
159. 1
199.5
158.4
126.9
102.8
0.0
0.0
104.9
120.7
141.4
167.8
141.2
129.2
102.3
0 .0
0.0
108.1
120.7
140.9
146.1
141.6
130.3
102.9
0.0
0.0
119.4
119.9
140.5
144.8
142.0
121.6
112.1
0.0
0.0
122.8
119.1.
140.5
144.5
141.5
120.9
117.8
0.0
0.0
128.6
116.9
141.0
144.2
141.6
119.9
120.2
0.0
0.0
129.0
116.2
140.7
144.4
141.8
117.3
123.7
0.0
0.0
127.6
114.6
141.2
144.1
141.7
116.8
124.0
0.0
0.0
122.6
114.1
141.2
144.1
141.7
115.9
121.1
0.0
0.0
117.3
113.0
141.3
144.0
142.1
114.3
118.0
0.0
0.0
82.9
111.5
142.9
144.9
143.4
113.7
87.1
0.0
0.0
79.5
107.8
143.1
144.7
143.8
110.2
83.9
0.0
0.0
71.6
101.4
138.4
141.8
139.3
103.7
75.6
0.0
0.0
0.0
51.5
61.9
77.8
90.3
127.3 111.5
131.5 116.8
128.3 112.5
79.8
92.5
65.5 55.3
0.0
0.0
0.0
42.4
65.0
95.1
99.7
96.1
66.4
46.1
0.0
0.0
33.5
52.7
78.1
83.4
79.2
54.4
36.5
0.0
0.0
25.4
40.3
61.7
66.4
62.7
41.7
27.8
0.0
17.1
28.5
44.7
50.0
45.8
29.7
18.9
0.0
0.0
0.0
9.5
16.7
28.4
33.0
29.4
17.7
10.7
0.0
0.0
3.1
7.9
14.1
17.3
14.7
8.6
3.8
0.0
0.0
0.2
2.2
4.5
5.7
4.7
2.6
0.5
0.0
0.0
0.0
0.0
0.6
0.9
0.6
0.1
0.0
0.0
37i. Trial with deposition:
deposition pattern at t=1440
PAGE 181
k. Description of deposition
The deposition pattern
deposition
pattern
that
would
nuclear explosion (see figure
The
be
9.105
likely
in
the
to result from a
Glasstone,
(1977)).
amount deposited grew very rapidly near the explosion and
quickly became a large, narrowly
cloud
realistically
fairly
reproduced
drifted
downwind,
distributed
peak.
As
the amount of particulate matter in
the cloud decreased and so the deposition rate was lower.
deposition
pattern
the
spread
downwind
The
across the grid but the
maximum amplitude that was attained grew smaller at increasing
distance from the explosion.
1. Cross sections through deposition pattern
Cross sections through the deposition pattern
below in figures 38a through 38c.
are
shown
PAGE 182
58.00
38a. Trial with deposition:
deposition cross-sections at t=0010
PAGE 183
28.00
33.00
LONGITUDE GRID#
38b. Trial with deposition:
deposition cross-sections at t=0100
PAGE 184
28.00
33.00
LONGITUDE GRID#
38c. Trial with deposition:
deposition cross-sections at t=1000
PAGE 185
The cross-sections illustrate the peak near the explosion
and
the
nearly exponential decay downwind.
The drop was not
exponential at the eastern portion of the pattern.
were
higher
than
an
exponential
decay would have produced
because they were produced by a cloud
truncation
error
deposited debris
eastern
portion
cross-section
particle
of
discussed
on
the
was
larger
because
1000.
The
than
At
enhancement
was
implied
timestep
flat
at
the
level
The deposition pattern
the.
of
the
by
the
1000,
the
eventually
that had already been reached at
gridpoints (35,26) through (37,26), where the cloud
passed.
of
cloud was over gridpoints (38,26) to (43,26) and was
depositing in that region.
became
that,
above, was not losing mass as it
ground.
timestep
The values
had
just
PAGE 186
V. Conclusions
A. Unsmoothed model
The Arakawa scheme in its pure form is not
the long term advection of particulate clouds.
developed with
during
lo ng
the
goal
that
conserved to within
attributed
were
tes ted
were
integration.
a
few
conserved
percent
as
quite
well.
monotonically to 1.7 to two times
origin
quantities
quantities,
Thes
well
fairly
the
total mass was al ways
with
error
the
The squared
The
its
mass
ea sily
was
not
quantity
always
grew
original
value.
The
of this growth has been discussed and the magnitud e of
the growth was not very large considering that the
two
in
to the trun cation errors that were in herent to the
integer arithmetic that was used.
conserved
The
here.
for
The scheme was
certai
conserving
term numerical
mass and squared mass,
trials
of
suitable
growth occurred over 1400 iterations.
factor
of
With more accurate
time differencing schem es, this quantity can be conserved more
closely, (Arakawa (1970
Unfortunately, these quantities
expense
of producing unrealistic results.
by the scheme tended to
cloud.
were
Indeed,
it
'was
almost
totally
conserved
at
the
The wakes produced
obscure
the
actual
difficult to even define the actual
PAGE 187
not
the
entire
cloud in these
trials.
distribution;
the negative concentrations are not realistic.
It
certainly
The positive area to the front of the
not,
itself
by
a
such
also
The mass and velocity of
this positive area did not agree with what should
in
was
distribution
actual cloud.
the
was
be
present
As pointed out in Orzag (1971), the wakes
cloud.
contained information about the distribution that
the
lagged
proper location of the cloud.
The entire distribution, including the wakes, produced by
the
pu re
had
scheme
a
centroid
that
was in the location
predict ed for particles moving at the loca 1 wind velocity.
likely th at this agreement is only preserved for cl ouds
seems
in
embedde d
present ed
ve ry
here,
idealized
the
wakes
where an actual cloud
concent rations
should
studied
never
here,
not
experienced.
the
field s.
In
tr ials
the
into regions of the grid
been
have
present.
The
thus advected by win ds that a real c loud
were
have
wind
spread
should
wakes
In
th e
simplified
c ases
always felt a pure zonal wind that
through the grid,
varied only very little
the
It
thus the locatio n of
centroid was not affected by the different winds that the
This would not, generally, be the case
wake was experiencing.
in the real atmosphere.
The pure scheme results in the extreme expansion
cloud.
above,
This
poses
but also is
problems
unre al istic
of
the
for the advection as described
in itself.
One of the ultimate
PAGE 188
goals
of
this
model
radioactive
particles
carried
the
by
spurious
predict the deposition rate of
falling
to
the
ground
after
being
of
the
cloud
implies
that
The
exposures
to
debris would occur over a much larger region than.
is realistic.
exposures
to
wind from an unwind nuclear detonation.
growth
radioactive
is
In particular, the
(
possibly
negative,
model
the
would
predict
that
meaning of which is an
enigma) even far upwind of the explosion.
This does not occur
in nature and is a major shortcoming of this model.
B. Smoothed model
In order to eliminate the
the
wakes,
a
with
This scheme was
very successful
at
wakes and restoring realistic results to the
the
advection scheme.
method
associated
smoothing scheme was introduced which modified
the pure Arakawa scheme.
controlling
difficulties
described
The wakes were totally
by
eliminated
the
in Appendix B, and the upwind growth of the
cloud was halted.
The clouds that were produced did exhibit some
diffusion,
it
but
was
greatly
reduced
and
positive.
The spatial extent of the clouds was
reduced.
The
numerical
exclusively
significantly
clouds were smaller than even just the forward
positive areas in
the
unsmoothed
trials.
The
most
rapid
PAGE 189
diffusion occurred during the early timesteps of the advect ion
that were initialized with *clouds covering only
grid
nera 11 y have
diffe renc ing schemes
vary o ver a sma
difficulty
obj ects
with
The minimum stable size in the
scale.
latitudinal direction appears to be five grid boxes .
one
and
nine
point
slow
growth
would
have
arithmetic had been used.
continued
It
pronounced,
very rapid after the initial timesteps.
expanded
only
likely
With integer arithmetic, very small
the directi on of motion was more
trials ,
is
the
if flo ating point
expansions were systematically rounded to zero.
the
When the
initializations grow to this width,
growth ceas ed for the remainder of the trial.
that
nine
or
This was not an unexpected result because fin ite
boxes.
that
one
to
Th e growth in
but
still
not
The clouds
in all of
and
thirteen
ten
between
gridpoints after over 1000 iterations.
his small
rate of numerical diffusion was entirely
positive
nd
not
diffusion,
he sum of the squared mass
Since
in
these
scheme.
Th
cloud.
predictions
counterbalanced
This
rials.
measure of
the
was
necessarily
decreased
major flaw of the smoothed
the
is
wake
by. negative
amount o f the decrease
of
this
is
quantity
a
he amount of numerical diffusion that has affected
problem
This
from
being
should
made.
not
preclude
useful
In nature, clouds experience
physical diffusion due to turbul ent motions in the atmosphere.
In
an
actual cloud, therefore, the sum of the squared masses
would drop.
In
the
'simplified
trials
presented
here
the
PAGE 190
natural
rate
of
diffusion
was
set
at
zero.
If natural
diffusion is to be incorporated into this model, the value
the
diffusion
constant
in (17)
for the numerical diffusion
of
should be reduced to account
already
present
in
the
model.
Thus a modified equation (17) would be:
[
DC
--
_
Dt
D
2
-D
]V (C)
E(natural) E(numeric) j
(37)
This smoothed model predicts very well the location of
particle
cloud
pressure.
In order
deposition
embedded
to
in
make
a
height
predictions
field
about
of
a
constant
the
actual
of radioactive debris, some additions must be made
to the scheme.
1. An appropriate diffusion constant must be chosen
This constant may be negative if the numerical
diffusion rate is determined to be larger than the
natural diffusion rate.
2. Depending on the type of debris that is under study,
a rate of decay should be incorporated into the model.
For many long lived isotopes, such as Strontium-90, this
decay rate can be safely ignored.
3. A modification should be made to the smoothing scheme,
if possible, to prevent the spurious lag in the velocity
PAGE 191
of the centroid of the distribution.
C. Deposition case
The
deposition
considering
the
trial
patterns
of
the
closely
that
well,
approxi mated
actuall, occur
especially
the
The
typical
in the aftermath of
The spurious e hanceme nt of
nuclear explosions.
portions
very
deposition scheme that was used.
crude
pattern that was produced
"teardrop"
worked
the
eastern
pattern could be easily eliminated with the
use of floating point , rather than integer, arithmetic in
the
calculation of the co ncentrations remaining in the cloud after
deposition.
If
considered,
then
onl y
a
long
more
range
deposition
assumes
terminal
which
it
with finite vertical extent,
veloci ty
the cloud falls it
(The
pressure levels are e asily
ground,
to
these wind s.)
until the
near
t
falls
the
In
at
at
the
level
to
height fields at several standard
obtainable
and
can
be
used
to
When the bottom of the cloud hits the
th en deposit i on begins to occur.
occur
be
of the particles contained in it. As
is advected by the wind
has fallen
cal cul ate
deposition
a vertical diffusion constant of zero.
this scheme the cloud
the
to
sophisticated scheme must be used.
The simplest scheme that eliminates the
explosion
is
Deposition continues
op of the cloud reaches the ground.
This
PAGE 192
scheme should produce a region of deposition
from
explosion
the
and
represents
that
solely
is
offset
long
range
deposition.
D. Summary
The model described here was tested on very idealized and
fields
simplified
of
wind
The physical
and concentration.
of
diffusion in the atmosphere was ignored and the decay rate
The deposition
radioactive particles was similarly neglected.
scheme was of the crudest nature.
A major simplification
made when the model incorporated only one layer.
was
This ignored
all vertical variation in the wind and concentration fields.
make
meaningful
predictions of actual fallout dispersion patterns.
Rather, it
The purpose of this study
was
not
to
was to develop and study the properties of a scheme which
be
used
to
make such predictions.
can
This paper discussed the
numeric instabilities of the advection scheme and the rates of
spurious
numerical
simulation of
serious
diffusion
fallout
dispersion.
are
to be expected in a
The
instabilities
were
but a smoothing scheme was presented which eliminated
the instabilities while
This
that
paper
should
still
provide
producing
realistic
results.
the groundwork for more detailed
studies of the long ra'nge dispersion of particles produced
by
PAGE 193
nuclear detonations.
PAGE 194
REFERENCES
Arakawa,
A. , (1966), "Computat ional Desi gn for Long-Term
Numerical Integration of Fluid Motions:
Incompressi ble Flow.
Part 1."
Two Dimensional
Journal of Computational
Physics, 1, 119-143.
Arakawa, A.,
(1970), "Numerical Simulation of Large-Scale
Atmospheric Motions", Numerical Solution of Field Problems
in Continuum Physics, (American Mathematical Society,
Providence, RI), 24-40
Arakawa A.,
and Lamb. V.', (1977), "Computational Design of
the Basic Dynamical Processes of the UCLA General
Circulation Model", Methods in Computational Physics,
Volume 17: General Circulation Models of the Atmosphere,
(Academic Press, New York, NY), 173-265
Bretherton, F.,
and Karweit, M .,
(1975), "Mid-Ocean Mesoscale
Modeling", Numerical Models of Ocean Circulations,
(National Academy of Sciences, Washington, DC), 237-249
Davidson, B.,
Friend, J . and Seitz,
H.,
(1966),
models of diffusion and rainout of stratospheric
radioactive materials",
Tellus,
18,
301-315
"Numerical
PAGE 195
Dyer, A.,
(1966), "Artificial radio-activity, ozone and
volcanic dust as atmospheric tracers in the Southern
Hemisphere", Tell us, 18, 416-419
Fjortoft,
R.,
(1953), "On the Changes in the Spectral
Distribution of Kinetic Energy for Twodimensional,
Nondivergent Flow
Glasstone, S.,
'",
Tellus, 5, 225-
and Dolan, P.,
(1977), The Effects of Nuclear
Weapons, (United States Department of Defense and United
States Department Qf Energy, Washington, DC)
Holton, J.,
(1979), An Introduction to Dynamic Meteorology,
Second Edition, (Academic Press, New York, NY)
Houghton, J.,
(1979), The Physics of Atmospheres,
(Cambridge University Press, Cambridge, United Kingdom)
Kao, S.-K., and Henderson, D.,
(1970), "Large-Scale
Dispersion of Clusters of Particles in Various Flow
Patterns", Journal of Geophysical Research, 75, 3104-3113
Lilly, D.,
(1965),
"On the Computational Stability of
Numerical Solutions of Time-Dependent Non-Linear
Geophysical Fluid Dynamics Problems", Monthly Weather
Review, 93, 11-26
PAGE 196
Lilly, D., (1969), "Numerical
Simulation of Two-Dimensional
Turbulence", The Physics of- Fluids Supplement, 2, 240-249
Mol enkamp,
C.,
(1967),
"Accuracy of Finite-Difference
Methods Applied to the Advection Equation",
Journal of
Applied Meteorology, 7, 160-167
Orzag,
(1971).
S.,
"Numerical simulation of incompressible
flows within simple boundaries:
Mechanics, 49,
Pedlosky,
accuracy", Journal of Fluid
75-112
(1979),
(Springer-Verlag,
Geophysical
Fluid Dynamics,
New York, NY)
Roache, (1976), Computational Fluid Dynamics,
Publishers,
Albuquerque;
Tanaevsky, 0.,
NM)
and Blanchet,
J.,
the course of radioactive debris",
Wit J.,
and Zagurek, M.,
(Hermosa
(1981),
MX", Arms Control Today, 11, 4-5
"Meteorological
Tellus,
"Fallout
18,
study of
434-439
and the Land-Based
PAGE 197
Appendix A: Height field generation scheme
The data used to construct the height field used
advection
calculation
is
transmitted
from
in
several hundred
observing locations across the North American continent.
observing
stations
are
across the continent.
field
the
The
scattered in a non-systematic manner
It is necessary to construct
a
height
from these observations so that there is a height value
defined for each gridpoint in the model.
The scheme presented here to construct a height field
is
not the only one which would produce acceptable height fields.
It is just one method of interpolating between
observations.
The
results
of
acceptable and agree well with other
analysis
of the same data.
this
other
methods,
the
scattered
interpolation
human
and
are
computerized
Where discrepancies arise between
the results produced by this method and the
by
the
differences
results
produced
can be explained in the
ambiguity of the field defined by the scattered
observations.
The mechanism of the interpolation is described below.
The scheme places all the scattered observations
grid.
It
then
finds
each of the observations.
these
pairs
of
in
the
a nearest neighboring observation for
Lines
observations.
are
drawn
between
all
of
The values of the heights on
these lines are linear interpolations between
the
endpoints.
PAGE 198
The grid then contains a network of lines that connect many of
the observations together.
Two interpolating passes are
In the first, the scheme searches for filled grid
conducted.
boxes with the same longitudinal
gri dpoint
inter polated lines between them.
linearly
most of a grid with height values.
this
proc ess,
but
gridpoint value
The
filled.
interpolat ed
contains
two
fo rms
ids
height
The
another
se con d
pass
repeats
grid
that
mostly
is
are then averaged to form the final
fiel d.
The
result,
at
stage,
this
discon tinuities in the first derivat ive of
between
drawn
In orde r to eliminate these discontin uities,
the averaged int erpola tion was smoothed.
chosen
and draws
Th is usually fills
the regio n of the lines that were
the observations
fiel d.
values
values with t he same latitudinal
between
This
equent
the field
was
The smoothing scheme
so as to el iminate short period variations in the
Each gri dpoint
val ue was modified to be
the
we ighted
aver age of itsel f and the neighboring gridpoint values.
cent ral
height
value
lo cated
at
gridpoint
(ln,lt),
following quantities were defined:
cc
dd
height(ln,1t)
=
( height(l n+1, lt)
+ height(ln-1,lt)
+ height( ln,lt+1) + height(ln,lt-1))/4
ee
=
then
( height(l n+1,lt+1)
+
height(ln+1,lt-1)
+ height( ln-1,lt+1) + height(ln-1,lt-1))/4
For a
the
PAGE 199
ff
=
( height(l n+2,lt) + height(ln-2,lt)
+ height( ln,lt+2) + height(ln,lt-2))/4
gg
=
( height(l n+2,lt+1) + height(ln+1,l t+2)
+ height( ln+2,lt-1) + height(ln+1, lt-2)
+ height( ln-2,1t+1) + height(ln-1, lt+2)
+ height( ln-2,lt-1) + height(ln-1, lt-2))/8
( height(l n+2 ,,1t+2)
hh
t-2)
+ height(l n+2,
+ height( ln-2, lt+2) + height( ln-2
The smoothed value at the gridpoint was the
of all these quantities.
lt-2 ))/4
weighted
average
The weighting function was the value
of a Gaussian normal curve with a stand rd deviation equal
the
between -two
distance
example, cc
weighted
by
was
weighted
normal(V17 +22
adjacent
by
normal(
)=.2136.
Thus, for
ridpoints.
)=.3989,
Using
to
and
gg
was
this weighting
function, the average was:
he-ight(ln, lt)=
.3989*cc+.3521*dd+.3108*ee+.2420*ff+.2136*gg+.1468*hh)
----------------------------------------------------(.3989+.3521+.3108+.2420+.2136+.1468)
This produced fields that were very smooth and represented the
large scale, synoptic,' variations in the actual height field.
PAGE 200
Appendix B:
Concentration field smoothing scheme
The problems presented by
detail
in
the
the
wakes
main body of this paper.
eliminating these wakes is presented
means
the
eliminating
only
possible,
these
wakes.
or
It
discussed
are
in
A simple scheme for
here.
This
even
the
best,
is,
however,
is
by
no
method
of
a
simple,
only
extend
straightforward method.
The effects of the Arakawa advection scheme
to
the
eight nearest neighbors of a grid box with a non-zero
Thus, in the initial wake forming
concentration.
wake
can
only
be
one
layer
thick.
shows, schematically, the region of
around
wake
possible
formation
The wind
around all grid boxes with concentrations.
sufficiently
advection
in
the
not
do
Wakes
right.
will not form if the concentrations in
the
The following diagram
each grid box with a positive concentration.
is blowing from the left to the
tested
phase,
form
The negative wakes
wake
region
are
positive so as to remain positive even after the
scheme
this
subtracts
paper,
concentration.
wakes
will
also
In
not
the
trials
form
if the
magnitude of the wake is truncated or rounded to zero
integer arithmetic used in the advection.
by
the
PAGE 201
beyond
range
0
cloud
0
poss.
wake
poss.
wake
poss.
cloud
wake
poss.
wake
0
0
0
0
timestep 0001
timestep 0000
Wake formation regions
Figure 39:
As the diagram shows, when a wake forms, all
containing
poss.
wake
wake
the
grid
boxes
particles are, in the first timestep of wake
formation, adjacent or
diagonally
adjacent
to
a
grid
box
occupied by cloud particles.
The wake el imination scheme eliminates the negative
timestep
every
and,
layer dimension.
thus, a wake never grows beyond the one
A secondary positive wake is never formed.
The method of elimination is very
scheme
searches
advection scheme .
searches
for
the
wake
straightforward.
fhe
for a negative concentration produced by the
When a negdtivc value is found, the
scheme
largest positive value that is adjacent or
PAGE 202
diagonally
adjacent
concentration
in
to
the
largest neighboring
the
negative
positive
value
and
it
The
possible
is
the
value
in
the
The negative value is added
to the largest nearby value instead of
because
box.
negative grid box is then added to the
negative grid box is set to zero.
neighbor
grid
the
absolute
nearest
that the absolute nearest
neighbor will be smaller in magnitude than the wake value.
An
addition would then result in just another negative value.
As
long as the CFL criteria is honored, there will
adjacent
or
diagonally
always
be
a
adjacent cloud concentration that is
larger than the wake concentration.
The
elimination
results
in
they occur.
he eliminat ion of
Trailers result when
isolates
which
particles,
were
in
concentration gradients, tha t were being left
advection
scheme.
In
nega tive
regions
of
low
behind
by
the
add ition to eliminati ng the wake, the
smoothing scheme made an att empt t
In
being
trailers
These are described in the particular trials in which
formed.
wakes
sometimes
eliminate these
trail ers.
the trials studied here, sever 1 different versions of the
smoothing
scheme
aggressiveness
version
of
were
in
eliminat ing
smoother
the
particles.
Other
quadruplets
of
version
use d.
versions
isolated
eliminated
only
differed
T hey
t he
eliminated
parti cles.
single
pairs,
The
t heir
The earl iest
trailers
e liminated
in
most
isol ated
triplets and
aggressive
long li nes of trai 1ers were only attached
to the bulk of the cl'ud by a one gridpoint
thick
string
of
PAGE 203
concentrations.
When found, a trailer was eliminated without
regard for conservation of mass.
The
concentrations
in the
trailers were small and this did not introduce any significant
errors.
The search for, and elimination of trailers
consuming
process
the advection.
the
was
a time
and was conducted only infrequently during
The short period oscillations in the
sum
of
squares of the area weighted concentration and squares of
concentrations are a result of trailers
the smoothing scheme.
being. eliminated
by